††thanks:  Keywords : Hardy’s Theorem; Bessel functions; Jacobi theta
function; confluent hypergeometric function 2020 Mathematics Subject
Classification : Primary: 11E45, 11M41, 33C10, 33C15; Secondary: 30B50, 44A15
Department of Mathematics, Faculty of Sciences of University of Porto, Rua do
Campo Alegre, 687; 4169-007 Porto (Portugal). E-mail:
<EMAIL_ADDRESS>
# Estimates for the number of zeros of shifted combinations of completed
Dirichlet series

Pedro Ribeiro

###### Abstract

In a previous paper [24], Yakubovich and the author of this article proved
that certain shifted combinations of completed Dirichlet series have
infinitely many zeros on the critical line. Here we provide some lower bounds
for the number of critical zeros of a subclass of shifted combinations.

###### Contents

  1. 1 Introduction and main results
  2. 2 Lemmas for the proof of Theorem 1.1
  3. 3 Proof of Theorem 1.1
    1. 3.1 Outline of the Proof
    2. 3.2 A suitable integral representation
    3. 3.3 Studying the sign changes of $\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$
    4. 3.4 The dominance of $\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$.
    5. 3.5 Conclusion of the argument
  4. 4 Lemmas for the proof of Theorem 1.2
  5. 5 Proof of Theorem 1.2
  6. 6 Concluding Remarks

##  1 Introduction and main results

Let $\eta(s):=\pi^{-s/2}\Gamma(s/2)\,\zeta(s)$. A. Dixit, N. Robles, A. Roy
and A. Zaharescu [8] proved the following theorem.

#### Theorem A:

Let $(c_{j})_{j\in\mathbb{N}}$ be a sequence of non-zeros real numbers such
that $\sum_{j=1}^{\infty}\,|c_{j}|<\infty$. Also, let
$\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ be a bounded sequence of distinct
real numbers that attains its bounds. Then the function

$F(s)=\sum_{j=1}^{\infty}c_{j}\,\eta\left(s+i\lambda_{j}\right):=\sum_{j=1}^{\infty}c_{j}\,\pi^{-\frac{s+i\lambda_{j}}{2}}\Gamma\left(\frac{s+i\lambda_{j}}{2}\right)\,\zeta\left(s+i\lambda_{j}\right)$

has infinitely many zeros on the critical line $\text{Re}(s)=\frac{1}{2}$.

See also the introduction of [8] for an excellent survey on results about
zeros of certain shifts of the Riemann zeta function. Based on an integral
representation of Jacobi’s transformation formula due to Dixit [[6], p. 374,
eq. (1.13)], A. Dixit, R. Kumar, B. Maji and A. Zaharescu [7] later
generalized the aforementioned result and proved the more general theorem.

#### Theorem B:

Let $(c_{j})_{j\in\mathbb{N}}$ and $(\lambda_{j})_{j\in\mathbb{N}}$ be as in
Theorem A. Also, let $\mathscr{R}$ denote the region of the complex plane
defined by
$\mathscr{R}:=\left\\{z\in\mathbb{C}\,:\,|\text{Re}(z)|<\sqrt{\frac{\pi}{2}},\,|\text{Im}(z)|<\sqrt{\frac{\pi}{2}}\right\\}$.
Then, for any $z\in\mathscr{R}$, the function

$F_{z}(s)=\sum_{j=1}^{\infty}c_{j}\,\eta\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{1-(s+i\lambda_{j})}{2};\,\frac{1}{2};\,\frac{z^{2}}{4}\right)\,+\,_{1}F_{1}\left(\frac{1-(\overline{s}-i\lambda_{j})}{2};\,\frac{1}{2};\,\frac{\overline{z}^{2}}{4}\right)\right\\}$
(1.1)

has infinitely many zeros on the critical line $\text{Re}(s)=\frac{1}{2}$.

The proof of Theorem B employed a variant of Hardy’s method of studying the
moments of the real function $\eta\left(\frac{1}{2}+it\right)$ [[14], [31],
Chapter X], as well as the elegant transformation formula

$\displaystyle 2x^{1/4}\psi(x,z)-x^{-1/4}e^{-z^{2}/4}$
$\displaystyle=2e^{-z^{2}/4}x^{-1/4}\,\psi\left(\frac{1}{x},\,iz\right)-x^{1/4}$
$\displaystyle=\frac{1}{2\pi}\,\intop_{-\infty}^{\infty}\pi^{-\frac{1}{4}-\frac{it}{2}}\Gamma\left(\frac{1}{4}+\frac{it}{2}\right)\,\zeta\left(\frac{1}{2}+it\right)$
$\,{}_{1}F_{1}\left(\frac{1}{4}+\frac{it}{2};\,\frac{1}{2};\,-\frac{z^{2}}{4}\right)\,x^{-\frac{it}{2}}\,dt,$
(1.2)

where

$\psi(x,z):=\sum_{n=1}^{\infty}e^{-\pi n^{2}x}\cos\left(\sqrt{\pi
x}\,n\,z\right),\,\,\,\,\text{Re}(x)>0,\,\,\,z\in\mathbb{C}.$ (1.3)

The first equality in (1.2) is, of course, due to Jacobi but the integral
representation appears for the first time in [[6], p. 374, eq. (1.13)].

Together with Yakubovich [24], the author of this paper extended Theorem B to
a class of Dirichlet series satisfying Hecke’s functional equation. A direct
generalization of Theorem B concerns Dirichlet series attached to positive
powers of the $\theta-$function,

$\theta(x):=\sum_{n=-\infty}^{\infty}e^{-\pi n^{2}x},\,\,\,\,\text{Re}(x)>0,$
(1.4)

defined as follows: for any $\alpha>0$, $r_{\alpha}(n)$ is the arithmetical
function [17] described by the expansion

$\theta^{\alpha}(x)-1:=\sum_{n=1}^{\infty}r_{\alpha}(n)\,e^{-\pi
nx},\,\,\,\,\,\text{Re}(x)>0.$ (1.5)

For $\text{Re}(s)$ sufficiently large, one may consider the Dirichlet series

$\zeta_{\alpha}(s):=\sum_{n=1}^{\infty}\frac{r_{\alpha}(n)}{n^{s}},$ (1.6)

and motivate its study through the transformation formula for $\theta(x)$.
When $\alpha=k\in\mathbb{N}$, it is effortless to see that $r_{\alpha}(n)$
reduces to the arithmetical function counting the number of representations of
$n$ as a sum of $k$ squares. Also, when $\alpha=1$, $\zeta_{1}(s)$ reduces to
$2\zeta(2s)$. Like $\zeta_{k}(s)$ and $\zeta(2s)$, $\zeta_{\alpha}(s)$
satisfies Hecke’s functional equation

$\eta_{\alpha}(s):=\pi^{-s}\Gamma(s)\,\zeta_{\alpha}(s)=\pi^{-\left(\frac{\alpha}{2}-s\right)}\Gamma\left(\frac{\alpha}{2}-s\right)\,\zeta_{\alpha}\left(\frac{\alpha}{2}-s\right):=\eta_{\alpha}\left(\frac{\alpha}{2}-s\right),$
(1.7)

from which it is possible to conclude that $\eta_{\alpha}(s)$ is real on the
critical line $\text{Re}(s)=\frac{\alpha}{4}$. The extension of Theorem B to
the class of zeta functions $\zeta_{\alpha}(s)$ is described by the following
result (cf. [[24], p. 6, Theorem 1.1]).

#### Theorem C:

Let $(c_{j})_{j\in\mathbb{N}}$ be a sequence of real numbers such that
$\sum_{j}|c_{j}|<\infty$ and $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ be a
bounded sequence of real numbers attaining its bounds. Then, for any $z$
satisfying the condition

$z\in\mathscr{D}_{\alpha}:=\left\\{z\in\mathbb{C}\,:\,|\text{Re}(z)|<\sqrt{\frac{\pi\alpha}{2}},\,|\text{Im}(z)|<\sqrt{\frac{\pi\alpha}{2}}\right\\},$
(1.8)

the function

$F_{z,\alpha}(s):=\sum_{j=1}^{\infty}c_{j}\,\pi^{-(s+i\lambda_{j})}\Gamma\left(s+i\lambda_{j}\right)\,\zeta_{\alpha}(s+i\lambda_{j})\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{\overline{z}^{2}}{4}\right)\right\\}$
(1.9)

has infinitely many zeros on the critical line
$\text{Re}(s)=\frac{\alpha}{4}$.

Note that Theorem B is a particular case of Theorem C (obtained when
$\alpha=1$). One of the main ingredients in the proof of Theorem C is a
generalization of Dixit’s integral formula (1.2) obtained in [[24], p. 30, eq.
(2.60)]. This identity takes the form

$\displaystyle
x^{\frac{\alpha}{4}}\psi_{\alpha}(x,z)-x^{-\alpha/4}\,e^{-\frac{z^{2}}{4}}$
$\displaystyle=e^{-\frac{z^{2}}{4}}x^{-\alpha/4}\,\psi_{\alpha}\left(\frac{1}{x},\,iz\right)-x^{\alpha/4}$
$\displaystyle=\frac{1}{2\pi}\,\intop_{-\infty}^{\infty}\eta_{\alpha}\left(\frac{\alpha}{4}+it\right)$
$\,{}_{1}F_{1}\left(\frac{\alpha}{4}+it;\,\frac{\alpha}{2};\,-\frac{z^{2}}{4}\right)\,x^{-it}\,dt,$
(1.10)

where $\eta_{\alpha}(s)$ is the completed Dirichlet series (1.7) and
$\psi_{\alpha}(x,z)$ is the generalization of Jacobi’s $\psi-$function,

$\psi_{\alpha}(x,z)=2^{\frac{\alpha}{2}-1}\,\Gamma\left(\frac{\alpha}{2}\right)\,\left(\sqrt{\pi
x}\,z\right)^{1-\frac{\alpha}{2}}\,\sum_{n=1}^{\infty}r_{\alpha}(n)\,n^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\pi
n\,x}\,J_{\frac{\alpha}{2}-1}(\sqrt{\pi\,n\,x}\,z),\,\,\,\,\,\text{Re}(x)>0,\,\,z\in\mathbb{C}.$
(1.11)

The transformation formula (1.10) can be also taken in a general setting, with
$\zeta_{\alpha}(s)$ being replaced by any Dirichlet series satisfying Hecke’s
functional equation. For example, let $f(\tau)$ be a holomorphic cusp form
with weight $k\geq 12$ for the full modular group whose Fourier expansion is
given by

$f(\tau)=\sum_{n=1}^{\infty}a_{f}(n)\,e^{2\pi
in\tau},\,\,\,\,\,\text{Im}(\tau)>0.$ (1.12)

If we construct the Dirichlet series associated to $f(\tau)$,

$L(s,f)=\sum_{n=1}^{\infty}\frac{a_{f}(n)}{n^{s}},\,\,\,\,\,\,\text{Re}(s)>\frac{k+1}{2},$
(1.13)

we know that $L(s,f)$ can be analytically continued to an entire function
obeying Hecke’s functional equation

$\eta_{f}(s):=(2\pi)^{-s}\Gamma(s)\,L(s,f)=(-1)^{k/2}\,\left(2\pi\right)^{-(k-s)}\,\Gamma(k-s)\,L(k-s,f):=(-1)^{k/2}\eta_{f}\left(k-s\right).$
(1.14)

Analogously to $\psi_{\alpha}(x,z)$, which generalizes Jacobi’s
$\psi-$function, we can construct a generalization of the cusp form $f(\tau)$
in the form111Note that, when $z=0$, $\psi_{f}(x,0)=f(ix)$

$\psi_{f}(x,z):=(k-1)!\,\left(\sqrt{\frac{\pi
x}{2}}\,z\right)^{1-k}\sum_{n=1}^{\infty}a_{f}(n)n^{\frac{1-k}{2}}\,e^{-2\pi
n\,x}\,J_{k-1}\left(\sqrt{2\pi
n\,x}\,z\right),\,\,\text{Re}(x)>0,\,\,z\in\mathbb{C}.$ (1.15)

Just like $\psi_{\alpha}(x,z)$, $\psi_{f}(x,z)$ obeys to the following
transformation

$\displaystyle x^{\frac{k}{2}}\psi_{f}(x,z)$
$\displaystyle=(-1)^{k/2}\,e^{-\frac{z^{2}}{4}}x^{-k/2}\,\psi_{f}\left(\frac{1}{x},\,iz\right)$
$\displaystyle=\frac{1}{2\pi}\,\intop_{-\infty}^{\infty}\left(2\pi\right)^{-\frac{k}{2}-it}\Gamma\left(\frac{k}{2}+it\right)\,L\left(\frac{k}{2}+it,f\right)$
$\,{}_{1}F_{1}\left(\frac{k}{2}+it;\,k;\,-\frac{z^{2}}{4}\right)\,x^{-it}\,dt$
$\displaystyle:=\frac{1}{2\pi}\,\intop_{-\infty}^{\infty}\eta_{f}\left(\frac{k}{2}+it\right)\,_{1}F_{1}$
$\displaystyle\left(\frac{k}{2}+it;\,k;\,-\frac{z^{2}}{4}\right)\,x^{-it}dt,$
(1.16)

where $\eta_{f}(s)$ is the completed Dirichlet series (1.14). Using (1.16), we
have been able to establish the following Theorem [[24], Theorem 1.4].

#### Theorem D:

Let $f(\tau)$ be a cusp form of weight $k$ for the full modular group with
real Fourier coefficients $a_{f}(n)$. Consider the Dirichlet series,

$L(s,f)=\sum_{n=1}^{\infty}\frac{a_{f}(n)}{n^{s}},\,\,\,\,\,\text{Re}(s)>\frac{k+1}{2}.$
(1.17)

If $(c_{j})_{j\in\mathbb{N}}$ is a sequence of non-zero real numbers such that
$\sum_{j=1}^{\infty}\,|c_{j}|<\infty$,
$\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ is a bounded sequence of distinct
real numbers222In the statement of Theorem 1.4. of [24] we require that the
sequence $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ attains its bounds.
However, in view of Remark 5.4 of [24], this condition is not necessary when
we are working with combinations involving entire Dirichlet series. and $z$
satisfies the condition

$z\in\mathscr{D}:=\left\\{z\in\mathbb{C}\,:\,|\text{Re}(z)|<2\sqrt{\pi},\,\,|\text{Im}(z)|<2\sqrt{\pi}\right\\},$
(1.18)

then the function

$G_{z,f}\left(s\right):=\sum_{j=1}^{\infty}c_{j}\,\left(2\pi\right)^{-s-i\lambda_{j}}\Gamma(s+i\lambda_{j})\,L\left(s+i\lambda_{j},f\right)\,\left\\{{}_{1}F_{1}\left(k-s-i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(k-\overline{s}+i\lambda_{j};\,k;\,\frac{\overline{z}^{2}}{4}\right)\right\\}$
(1.19)

has infinitely many zeros at the critical line $\text{Re}(s)=\frac{k}{2}$.

Our main goal in this paper is to prove quantitative analogues of Theorems C
and D above for a subclass of shifted combinations. This is, when we impose
some further restrictions on $(\lambda_{j})_{j\in\mathbb{N}}$, we aim to find
lower bounds for the number of critical zeros of the functions
$F_{z,\alpha}(s)$ and $G_{z,f}(s).$ We need to introduce the following
assumptions:

  1. 1.

Assumpion 1: If the shift $s\rightarrow s+i\lambda_{j}$ is introduced, then
the symmetric shift $s\rightarrow s-i\lambda_{j}$ needs to be introduced as
well and having the same weight $c_{j}$. Concerning Theorem C, for example,
this means that if the term

$c_{j}\,\eta_{\alpha}\left(s+i\lambda_{j}\right)\text{Re}\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right\\}$

belongs to the combination (1.9), then the term

$c_{j}\,\eta_{\alpha}\left(s-i\lambda_{j}\right)\text{Re}\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right\\}$

also belongs to it. This assumption essentially says that we are enlarging the
sequences $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ in (1.9) and (1.19) in
such a way that each element has a symmetric pair contained in it. Therefore,
we can write it as a sequence over $\mathbb{Z}\setminus\\{0\\}$ in the form
$\left(\lambda_{j}\right)_{j\in\mathbb{Z}\setminus\\{0\\}}=\left(\lambda_{j}\right)_{j\in\mathbb{N}}\cup\left(-\lambda_{j}\right)_{j\in\mathbb{N}}$,
once we take the convention that $\lambda_{-j}:=-\lambda_{j}$. The same can be
done to $(c_{j})_{j\in\mathbb{N}}$ under the convention $c_{-j}:=c_{j}$.

  2. 2.

Assumption 2: In each one of the cases (1.9) and (1.19), the parameter $z$ in
the hypergeometric functions will be a real number belonging to an interval
contained in the regions (1.8) and (1.18).

Under these assumptions, we can write the shifted combinations (1.9) and
(1.19) in the symmetric forms

$\tilde{F}_{z,\alpha}(s):=\sum_{j\neq
0}c_{j}\,\pi^{-(s+i\lambda_{j})}\Gamma\left(s+i\lambda_{j}\right)\,\zeta_{\alpha}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right\\}$
(1.20)

and

$\tilde{G}_{z,f}(s):=\sum_{j\neq
0}c_{j}\,\left(2\pi\right)^{-s-i\lambda_{j}}\Gamma(s+i\lambda_{j})\,L\left(s+i\lambda_{j},f\right)\,\left\\{{}_{1}F_{1}\left(k-s-i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(k-\overline{s}+i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)\right\\}.$
(1.21)

In both cases, the sum is taken over $\mathbb{Z}\setminus\\{0\\}$ and we are
under the conventions (see Assumption 1) $c_{-j}:=c_{j}$ and
$\lambda_{-j}:=-\lambda_{j}$.

Note also that $\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)$ is a
real-valued and even function of $t\in\mathbb{R}$. These properties come
immediately from the functional equation (1.7) and assumptions 1 and 2. In the
same lines one can check that, when the coefficients $a_{f}(n)$ are real,
$i^{-\frac{k}{2}}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)$ is a real
function of $t$. Furthermore, it is an even function if $k\equiv 0\mod 4$ and
it is odd if $k\equiv 2\mod 4$. Although the additional assumptions given
above restrict some of the general aspects of Theorems C and D, they prove to
be very helpful in deriving some estimates for the number of zeros of the
functions (1.20) and (1.21). We begin with a theorem about the number of zeros
of $\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)$.

######  Theorem 1.1.

Let $(c_{j})_{j\in\mathbb{N}}$ be a sequence of real numbers such that
$\sum_{j=1}^{\infty}|c_{j}|<\infty$ and $(\lambda_{j})_{j\in\mathbb{N}}$ be a
bounded sequence of real numbers attaining its bounds. Suppose that
$(c_{j})_{j\in\mathbb{N}}$ and $(\lambda_{j})_{j\in\mathbb{N}}$ can be
extended to $\mathbb{Z}\setminus\\{0\\}$ in the form $c_{-j}=c_{j}$ and
$\lambda_{-j}=-\lambda_{j}$. Assume also that $z\in\mathbb{R}$ satisfies the
condition

$z\in\left[-\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}},\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}}\right].$
(1.22)

Moreover, let $N_{\alpha,z}(T)$ be the number of zeros written in the form
$s=\frac{\alpha}{4}+it,\,\,\,0\leq t\leq T$, of the function

$\tilde{F}_{z,\alpha}(s):=\sum_{j\neq
0}c_{j}\,\pi^{-(s+i\lambda_{j})}\Gamma\left(s+i\lambda_{j}\right)\,\zeta_{\alpha}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right\\}.$
(1.23)

Then there exists some $c>0$ such that

$\liminf_{T\rightarrow\infty}\,\frac{N_{\alpha,z}(T)}{\sqrt{T}/\log(T)}\geq
c.$ (1.24)

The quantitative estimate (1.24) is somewhat general and it seems very
difficult to improve on the power of $T$ using the same method. In the final
section of this paper, we make some conjectures regarding a possible
improvement. When the shifted combination is trivial and reduces to the term
$\lambda_{1}=0$, our estimate (1.24) extends, for all $\alpha>0$, a result of
the author and Yakubovich [[25], Corollary 3.5] (cf. Remark 1.1 below).

Our proof of Theorem 1.1 employs a method developed by Fekete [10], which was
essentially inspired by a lemma of Fejér [9]. See [[31], p. 259] for a clear
explanation of Fekete’s method, as well as an interesting paper by Berlowitz
which uses the same idea [[3], p. 206, Lemma 2]. However, in order to apply
Fejér’s lemma, it will be crucial in our argument to use the fact that
$\zeta_{\alpha}(s)$ has a simple pole located at $s=\frac{\alpha}{2}$.

Since $L(s,f)$ is an entire function of $s$, a quantitative analogue of
Theorem D will have to use a different idea. In any case, we have been able to
establish the following result.

######  Theorem 1.2.

Let $f(\tau)$ be a cusp form of weight $k$ for the full modular group with
real Fourier coefficients $a_{f}(n)$ and consider the Dirichlet series,
$L(s,f)$, attached to it. Let $(c_{j})_{j\in\mathbb{N}}$ be a sequence of non-
zero real numbers such that $\sum_{j=1}^{\infty}\,|c_{j}|<\infty$ and let
$(\lambda_{j})_{j\in\mathbb{N}}$ be a bounded sequence of distinct real
numbers. Suppose that $(c_{j})_{j\in\mathbb{N}}$ and
$(\lambda_{j})_{j\in\mathbb{N}}$ can be extended to
$\mathbb{Z}\setminus\\{0\\}$ in the form $c_{-j}=c_{j}$ and
$\lambda_{-j}=-\lambda_{j}$. Assume also that $z$ is a real number satisfying
the condition

$z\in\left[-\frac{\sqrt{\pi}}{3},\frac{\sqrt{\pi}}{3}\right].$ (1.25)

If $N_{f,z}(T)$ denotes the number of zeros written in the form
$s=\frac{k}{2}+it,\,\,0\leq t\leq T$, of the function

$\tilde{G}_{z,f}(s):=\sum_{j\neq
0}c_{j}\,\left(2\pi\right)^{-s-i\lambda_{j}}\Gamma(s+i\lambda_{j})\,L\left(s+i\lambda_{j},f\right)\,\left\\{{}_{1}F_{1}\left(k-s-i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(k-\overline{s}+i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)\right\\},$
(1.26)

then there exists some $d>0$ such that

$\limsup_{T\rightarrow\infty}\frac{N_{f,z}(T)}{\sqrt{T}}\geq d.$ (1.27)

Our proof of Theorem 1.2 uses a variant of a method due to de la Vallée
Poussin [5]. The author of this paper has recently used this method to
establish a quantitative estimate for the number of critical zeros of
$L-$functions attached to half-integral weight cusp forms [23].

This paper is organized as follows. In the next section we give the necessary
technical lemmas to establish Theorem 1.1. The most important of these, Lemma
2.2, is obtained via a generalization of the theta transformation formula
obtained in [24]. Section 3 is devoted to a proof of Theorem 1.1. Next, we
follow a similar structure in sections 4 and 5. Finally, we end this paper
with some conjectures concerning further extensions of the main results here
presented. Before moving on, we introduce a couple of remarks which describe
some interesting particular cases of our Theorems 1.1 and 1.2, as well as
other theorems that can be proved via the same methods.

######  Remark 1.1.

The intervals (1.22) and (1.25) are respectively contained in the regions
(1.8) and (1.18) of Theorems C and D. As we shall see below, it is possible to
enlarge the length of the interval (1.22) by a more careful choice of the
parameter $\lambda$ in the proof of Lemma 2.2 and a more precise inequality
for $|I_{\nu}(x)|$. See Remark 2.1 below.

######  Remark 1.2.

When $z=0$, we deduce from Theorem 1.1 that the arbitrary shifted combination
given by

$\sum_{j\neq
0}c_{j}\,\pi^{-(s+i\lambda_{j})}\Gamma\left(s+i\lambda_{j}\right)\,\zeta_{\alpha}\left(s+i\lambda_{j}\right)$

has $\gg T^{1/2}/\log(T)$ zeros of the form $s=\frac{\alpha}{4}+it,$ $0\leq
t\leq T$. Since $\zeta_{1}(s):=2\zeta(2s)$, this shows that the function

$F(s):=\sum_{j\neq
0}c_{j}\,\pi^{-\frac{1}{2}(s+i\lambda_{j})}\Gamma\left(\frac{s+i\lambda_{j}}{2}\right)\,\zeta\left(s+i\lambda_{j}\right)$
(1.28)

has $\gg T^{1/2}/\log(T)$ on the critical line $\text{Re}(s)=\frac{1}{2}$.
Also, when $z\neq 0$ and $\alpha=1$, Theorem 1.1 yields an extension of
Theorem B above. The generality of this result, with respect to the shifted
combination and the explicit estimate for the number of critical zeros, seems
to be unnoticed in the literature. We should remark that Selberg’s outstanding
result about the positive proportion of combinations of $L-$functions with
degree one [26] does not seem to cover the case (1.28), because we are
considering shifted combinations of $\eta(s)$.

######  Remark 1.3.

Using an inductive method involving generalized Epstein zeta functions [25],
Yakubovich and the author of this paper proved that, when $\alpha>4$, there
always exist $T_{0}(\alpha)$ and $c:=c(\alpha)>0$ such that, for any $T\geq
T_{0}(\alpha)$, there is a zero $\rho=\frac{\alpha}{4}+i\gamma$ of
$\zeta_{\alpha}(s)$ with $\gamma\in[T,T+cT^{1/2}\log(T)]$. See [[25],
Corollary 3.5] for details. In particular, if $N_{\alpha}(T):=\\#\left\\{0\leq
t\leq T\,:\,\zeta_{\alpha}\left(\frac{\alpha}{4}+it\right)=0\right\\}$, then

$\liminf_{T\rightarrow\infty}\,\frac{N_{\alpha}(T)}{\sqrt{T}/\log(T)}\geq
c^{\prime}>0,\,\,\,\,\,\alpha>4.$ (1.29)

Our Theorem 1.1 above generalizes the estimate (1.29) to shifted combinations
of $\zeta_{\alpha}(s)$. Moreover, when $z=0$ and $c_{1}=1$, $\lambda_{1}=0$,
$c_{2}=c_{3}=...=0$, it actually extends (1.29) also to the range
$0<\alpha\leq 4$. As remarked in [25], when $\alpha$ is any integer greater
than $3$, it is known that $N_{\alpha}(0,T)\asymp T$ [28] and so, in this
case, (1.29) does not say anything new. However, for non-integral $\alpha$,
this result seems to be novel.

######  Remark 1.4.

Analogues of our Theorem 1.1 can be proved when $\zeta_{\alpha}(s)$ is
replaced by any of the meromorphic Dirichlet series mentioned in [24]. For
example, it is still valid if we replace $\zeta_{\alpha}(s)$ by an Epstein
zeta function $\zeta(s,Q)$ attached to an integral quadratic form
$Q(m,n)=Am^{2}+Bmn+Cn^{2}$ such that $\sqrt{4AC-B^{2}}\equiv 2\mod 4$. Of
course, the admissible region for $z$ in this case would have to depend on the
discriminant of $Q$.

######  Remark 1.5.

A lower bound for $d$ appearing in (1.27) can be explicitly calculated. A
simple numerical computation (see the proof of Lemma 4.2 below) gives the
value $d=\frac{1}{36\pi}$ as admissible for $d$.

######  Remark 1.6.

Since the method of proof of Theorem 1.2 works well for any entire Dirichlet
series, there is an analogue of this result for some Dirichlet $L-$functions.
See [[24], Theorem 1.3] for details. Furthermore, we can use this method to
prove (1.27) for a general combination involving shifts of
$\zeta_{\alpha}(s)$. Since $s\left(s-\frac{\alpha}{2}\right)\eta_{\alpha}(s)$
is an entire function of $s$, then our proof of Theorem 1.2 can be applied to
study the number of zeros of the function

$\sum_{j\neq
0}c_{j}\left(s+i\lambda_{j}\right)\left(s+i\lambda_{j}-\frac{\alpha}{2}\right)\,\eta_{\alpha}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{{z}^{2}}{4}\right)\right\\},$
(1.30)

whenever $z$ satisfies (1.22). If $\tilde{N}_{\alpha,z}(T)$ denotes the number
of zeros of the above combination that can be written in the form
$s=\frac{\alpha}{4}+it,$ $0\leq t\leq T$, then the following estimate takes
place

$\limsup_{T\rightarrow\infty}\,\frac{\tilde{N}_{\alpha,z}(T)}{\sqrt{T}}\geq\tilde{c},$
(1.31)

for some $\tilde{c}>0$. In particular, when we reduce the combination (1.30)
to the case where $\lambda_{1}=0$ and $c_{j}=0$ for any $|j|\geq 2$, we see
that, besides (1.29), we also have
$N_{\alpha}(T)=\Omega\left(T^{\frac{1}{2}}\right)$.

######  Remark 1.7.

Using the properties of the slash operator, it is possible to establish
Theorem 1.2 when $f(z)$ is a cusp form of weight $k$ on a congruence subgroup
$\Gamma_{0}(N)$ with $N$ being a perfect square. We have to put the additional
condition that $\left(f|W_{N}\right)(z)=\pm f(z)$, where $W_{N}$ is the Fricke
involution. The same method also works for half-integral weight cusp forms in
$\Gamma_{0}(4N)$, $N$ being a perfect square.

##  2 Lemmas for the proof of Theorem 1.1

We start by recalling Stirling’s formula for the Gamma function,

$\Gamma(\sigma+it)=(2\pi)^{\frac{1}{2}}\,t^{\sigma+it-\frac{1}{2}}\,e^{-\frac{\pi
t}{2}-it+\frac{i\pi}{2}(\sigma-\frac{1}{2})}\left(1+\frac{1}{12(\sigma+it)}+O\left(\frac{1}{t^{2}}\right)\right),\,\,\,\,t\rightarrow\infty,$
(2.1)

valid whenever $-\infty<\sigma_{1}\leq\sigma\leq\sigma_{2}<\infty$. A similar
formula can be written for $t<0$ as $t$ tends to $-\infty$ by using the fact
that $\Gamma(\overline{s})=\overline{\Gamma(s)}$. Of course, a direct
consequence of this exact version is

$|\Gamma(\sigma+it)|=(2\pi)^{\frac{1}{2}}\,|t|^{\sigma-\frac{1}{2}}\,e^{-\frac{\pi}{2}|t|}\left(1+O\left(\frac{1}{|t|}\right)\right),\,\,\,\,\,|t|\rightarrow\infty.$
(2.2)

Let us now recall some basic facts about the Dirichet series
$\zeta_{\alpha}(s)$. It is well-known that the theta function
$\vartheta_{3}(\tau):=\sum_{n\in\mathbb{Z}}e^{\pi in^{2}\tau}$ is a modular
form of weight $\frac{1}{2}$ with a multiplier system with respect to the
theta group $\Gamma_{\theta}$ (see [17], p. 15-16 for details). Therefore, for
any $\alpha>0$, $\vartheta_{3}^{\alpha}(\tau)$ is a modular form of weight
$\alpha/2$ with a multiplier system on the same group. By definition,
$r_{\alpha}(n)$ are the Fourier coefficients of the expansion of
$\vartheta_{3}(\tau)$ at the cusp $i\infty$, this is (cf. [24] and [17] for a
clearer explanation of this expansion)

$\vartheta_{3}^{\alpha}(ix):=\theta^{\alpha}(x)=1+\sum_{n=1}^{\infty}r_{\alpha}(n)\,e^{-\pi
n\,x},\,\,\,\,\,x>0.$ (2.3)

The order of growth of $r_{\alpha}(n)$ as $n\rightarrow\infty$ is determined
by classical estimates due to Petersson and Lehner [17, 18]. These estimates
show that

$r_{\alpha}(n)\ll_{\alpha}\begin{cases}n^{\alpha/2-1}&\alpha>4\\\
n^{\alpha/2-1}\log(n)&\alpha=4\\\ n^{\alpha/4}&0<\alpha<4.\end{cases}$ (2.4)

Thus, we can rigorously define the Dirichlet series (1.6) in the form

$\zeta_{\alpha}(s):=\sum_{n=1}^{\infty}\frac{r_{\alpha}(n)}{n^{s}},\,\,\,\,\,\,\text{Re}(s)>\sigma_{\alpha}:=\begin{cases}\frac{\alpha}{2}&\alpha\geq
4\\\ 1+\frac{\alpha}{4}&0<\alpha<4\end{cases}.$ (2.5)

Mimicking Riemann’s paper, one can easily show that $\zeta_{\alpha}(s)$ can be
analytically continued to the entire complex plane as a meromorphic function
with a simple pole located at $s=\frac{\alpha}{2}$, whose residue is
$\text{Res}_{s=\alpha/2}\zeta_{\alpha}(s)=\frac{\pi^{\alpha/2}}{\Gamma(\alpha/2)}$.
Moreover, it satisfies Hecke’s functional equation

$\eta_{\alpha}(s):=\pi^{-s}\Gamma(s)\,\zeta_{\alpha}(s)=\pi^{-\left(\frac{\alpha}{2}-s\right)}\Gamma\left(\frac{\alpha}{2}-s\right)\,\zeta_{\alpha}\left(\frac{\alpha}{2}-s\right):=\eta_{\alpha}\left(\frac{\alpha}{2}-s\right).$
(2.6)

It also follows from the Phragmén-Lindelöf principle that $\zeta_{\alpha}(s)$
obeys to the convex estimate

$\zeta_{\alpha}\left(\sigma+it\right)\ll_{\alpha}|t|^{\sigma_{\alpha}-\sigma+\delta},\,\,\,\,\frac{\alpha}{2}-\sigma_{\alpha}-\delta<\sigma<\sigma_{\alpha}+\delta,$
(2.7)

for any $\delta>0$ and $\sigma_{\alpha}$ defined by (2.5).

In order to estimate the series given in (1.20), we need an asymptotic formula
for the confluent hypergeometric function valid when $|s|\rightarrow\infty$.
Following the reasoning in [[6], p. 379], recall that the Whittaker function
$M_{\lambda,\mu}(z)$ has the asymptotic formula, [[19], p. 341, eq. (13.21.1)]

$M_{\lambda,\mu}(z)=\frac{z^{1/4}}{\sqrt{\pi}}\lambda^{-\mu-\frac{1}{4}}\Gamma\left(2\mu+1\right)\,\cos\left(2\sqrt{\lambda
z}-\frac{\pi}{4}-\mu\pi\right)+O\left(|\lambda|^{-\mu-\frac{3}{4}}\right)$
(2.8)

as $|\lambda|\rightarrow\infty$ and $z$ such that $|\arg(\lambda z)|<2\pi$.
Furthermore, since

$M_{\lambda,\mu}(z)=z^{\mu+\frac{1}{2}}e^{-z/2}\,_{1}F_{1}\left(\mu-\lambda+\frac{1}{2};\,2\mu+1;\,z\right),$
(2.9)

we see that, as $|t|\rightarrow\infty$ and fixed $z\in\mathbb{R}$, the
substitutions in (2.8) and (2.9) give the bound

$\left|{}_{1}F_{1}\left(\frac{\alpha}{4}-it;\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right|=\left(\frac{|z|}{2}\right)^{-\frac{\alpha}{2}}e^{\frac{z^{2}}{8}}\left\\{\Gamma\left(\frac{\alpha}{2}\right)\sqrt{\frac{|z|}{2\pi}}\,|t|^{\frac{1-\alpha}{4}}\,\exp\left(\sqrt{\frac{|t|}{2}}\,|z|\right)+O\left(|t|^{-\frac{\alpha}{4}-\frac{1}{4}}\right)\right\\},\,\,\,\,|t|\rightarrow\infty.$
(2.10)

This estimate is more than enough to justify most of the steps in this paper.
By Stirling’s formula and (2.7), one can see that

$\left|\eta_{\alpha}\left(\frac{\alpha}{4}+it\right)\right|\ll_{\alpha}\,|t|^{A(\alpha)}\,e^{-\frac{\pi}{2}|t|},\,\,\,\,\,|t|\rightarrow\infty,$
(2.11)

where $A(\alpha)=\sigma_{\alpha}-\frac{1}{2}$, $\sigma_{\alpha}$ being given
by (2.5). When combined with (2.10), the convex estimate (2.11) yields the
bound

$\left|\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\right|\leq\sum_{j\neq
0}\left|c_{j}\,\eta_{\alpha}\left(\frac{\alpha}{4}+i\,(t+\lambda_{j})\right)\text{Re}\left(\,{}_{1}F_{1}\left(\frac{\alpha}{4}-i\,(t+\lambda_{j});\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right)\right|\ll_{\alpha,z}C_{\lambda}\sum_{j=1}^{\infty}|c_{j}|\,|t|^{B(\alpha)}\,e^{-\frac{\pi}{2}|t|+|z|\sqrt{|t|}},$
(2.12)

where we have used the fact that $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$
is a bounded sequence and $\left(c_{j}\right)_{j\in\mathbb{N}}\in\ell^{1}$.
The term $C_{\lambda}$ only stands for a positive constant which depends on
the bounds of the sequence $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$.

Next, let us note that an explicit way of writing the first equality in (1.2)
is

$\displaystyle
1+2^{\frac{\alpha}{2}-1}\,\Gamma\left(\frac{\alpha}{2}\right)\,\left(\sqrt{\pi
x}\,z\right)^{1-\frac{\alpha}{2}}\,\sum_{n=1}^{\infty}r_{\alpha}(n)\,n^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\pi
n\,x}\,J_{\frac{\alpha}{2}-1}\left(\sqrt{\pi n\,x}\,z\right)$
$\displaystyle=\frac{e^{-\frac{z^{2}}{4}}}{x^{\alpha/2}}\left\\{1+2^{\frac{\alpha}{2}-1}\Gamma\left(\frac{\alpha}{2}\right)\,\left(\sqrt{\frac{\pi}{x}}\,z\right)^{1-\frac{\alpha}{2}}\sum_{n=1}^{\infty}r_{\alpha}(n)\,n^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\frac{\pi
n}{x}}\,I_{\frac{\alpha}{2}-1}\left(\sqrt{\frac{\pi
n}{x}}\,z\right)\right\\},$ (2.13)

whenever $\text{Re}(x)>0$ and $z\in\mathbb{C}$. Since the summation formula
(2.13) transforms a generalized $\psi-$function involving the Bessel functions
of the first kind, it will be useful in our next argument to have a bound for
the modified Bessel function, $I_{\nu}(z)$, $\nu>-1$. In the previous paper
[[24], p. 42] we have used the famous Hankel expansion for $I_{\nu}(z)$,
$|z|\rightarrow\infty$. However, for our alternative argument we need a bound
for $|I_{\nu}(z)|$ that is somewhat uniform in $\nu$ and $z$.

The following simple bound will be useful: first, let us note that, if
$\nu\geq 0$ and $k\in\mathbb{N}_{0}$,

$\Gamma(k+\nu+1)=\Gamma(\nu+1)(\nu+1)...(\nu+k)\geq k!\,\Gamma(\nu+1),$

which gives, after the use of the power series of $I_{\nu}(z)$,

$|I_{\nu}(z)|\leq\left(\frac{|z|}{2}\right)^{\nu}\,\sum_{k=0}^{\infty}\frac{|z|^{2k}}{2^{2k}k!\,\Gamma(k+\nu+1)}\leq\frac{(|z|/2)^{\nu}}{\Gamma(\nu+1)}\,\sum_{k=0}^{\infty}\frac{|z|^{2k}}{2^{2k}(k!)^{2}}\leq\frac{(|z|/2)^{\nu}}{\Gamma(\nu+1)}\left\\{\sum_{k=0}^{\infty}\frac{|z|^{k}}{2^{k}k!}\right\\}^{2}=\left(\frac{|z|}{2}\right)^{\nu}\frac{e^{|z|}}{\Gamma(\nu+1)},\,\,\,\,\,\nu\geq
0.$ (2.14)

If, on the other hand, we want to extend (2.14) to $-1<\nu<0$, we begin to
note that, for $k\geq 1$,

$\Gamma(k+\nu+1)=\Gamma(\nu+2)\,(\nu+2)...(\nu+k)\geq(k-1)!\,\Gamma(\nu+2).$

Hence, if $-1<\nu<0$,

$|I_{\nu}(z)|\leq\left(\frac{|z|}{2}\right)^{\nu}\,\sum_{k=0}^{\infty}\frac{|z|^{2k}}{2^{2k}k!\,\Gamma(k+\nu+1)}\leq\frac{\left(|z|/2\right)^{\nu}}{\Gamma(\nu+1)}+\frac{\left(|z|/2\right)^{\nu+2}}{\Gamma(\nu+2)}\,\sum_{k=0}^{\infty}\frac{|z|^{2k}}{2^{2k}(k!)^{2}}<\left(\frac{|z|}{2}\right)^{\nu}\frac{e^{|z|}}{\Gamma(\nu+2)}\left(1+\frac{|z|^{2}}{4}\right).$
(2.15)

To proceed, let us recall some functions and facts already mentioned at the
introduction. From this point on, we will let $\alpha>0$ and $r_{\alpha}(n)$
be defined as the coefficients of the $q-$expansion of $\theta^{\alpha}(x)$.
For $\text{Re}(x)>0$ and $z\in\mathbb{C}$, we let $\psi_{\alpha}(x,z)$ denote
the analogue of Jacobi’s $\psi-$function,

$\psi_{\alpha}(x,z)=2^{\frac{\alpha}{2}-1}\,\Gamma\left(\frac{\alpha}{2}\right)\,\left(\sqrt{\pi
x}\,z\right)^{1-\frac{\alpha}{2}}\,\sum_{n=1}^{\infty}r_{\alpha}(n)\,n^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\pi
n\,x}\,J_{\frac{\alpha}{2}-1}(\sqrt{\pi\,n\,x}\,z).$ (2.16)

As stated above, the transformation formula for $\psi_{\alpha}(x,z)$, (2.13),
will play a major role in this paper. We shall also need some auxiliary
summation formulas given in [[24], pp. 37-40]. Analogously to $r_{\alpha}(n)$,
we can consider positive powers of the theta function

$\vartheta_{2}(\tau)=2\,\sum_{n=0}^{\infty}e^{\pi
i\tau\left(n+\frac{1}{2}\right)^{2}},\,\,\,\,\text{Im}(\tau)>0,$ (2.17)

from the Fourier expansion at the cusp $\tau=i\infty$. We can create a new
arithmetical function $\tilde{r}_{\alpha}(m)$ as the coefficient coming from
the expansion (see [[24], p.36, eq. (3.5)] for details)

$\displaystyle\vartheta_{2}^{\alpha}(ix)$
$\displaystyle:=\theta_{2}^{\alpha}(x)=\left(2\,\sum_{n=0}^{\infty}e^{-\pi
x\left(n+\frac{1}{2}\right)^{2}}\right)^{\alpha}=\left(2\,e^{-\frac{\pi
x}{4}}+2\,\sum_{n=1}^{\infty}e^{-\pi x(n+\frac{1}{2})^{2}}\right)^{\alpha}$
$\displaystyle=2^{\alpha}e^{-\frac{\pi\alpha
x}{4}}\left(1+\,\sum_{n=1}^{\infty}e^{-\pi
x\left(n^{2}+n\right)}\right)^{\alpha}=2^{\alpha}\,e^{-\frac{\pi\alpha
x}{4}}\,\sum_{j=0}^{\infty}\left(\begin{array}[]{c}\alpha\\\
j\end{array}\right)\,\left(\sum_{n=1}^{\infty}e^{-\pi
x\left(n^{2}+n\right)}\right)^{j}$ (2.20)
$\displaystyle:=\sum_{m=0}^{\infty}\tilde{r}_{\alpha}(m)\,e^{-\pi\left(m+\frac{\alpha}{4}\right)x},\,\,\,\,\,\,\,\,\,\,\,\text{Re}(x)>0.$
(2.21)

Analogously to $r_{\alpha}(n)$, the coefficients $\tilde{r}_{\alpha}(n)$ grow
polynomially with $n$ (cf.[[24], Lemma 3.1]). Note also that
$\tilde{r}_{\alpha}(0):=2^{\alpha}$ by the construction (2.21). Using Jacobi’s
transformation formula for $\vartheta_{2}(\tau)$, one can derive a summation
formula connecting $\tilde{r}_{\alpha}(n)$ with $r_{\alpha}(n)$. The next
lemma, given in [[24], p. 40, Lemma 3.3], establishes this correspondence.

######  Lemma 2.1 ([24], Lemma 3.3).

Let $r_{\alpha}(n)$ be the coefficients of the series expansion of
$\theta^{\alpha}(x)-1$, (1.5), and $\tilde{r}_{\alpha}(n)$ be defined by
(2.21). Then, for $\text{Re}(x)>0$ and $y\in\mathbb{C}$, the following
identity holds

$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\,r_{\alpha}(n)\,n^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\pi
nx}\,J_{\frac{\alpha}{2}-1}(y\,\sqrt{\pi\,n})$
$\displaystyle=-\frac{y^{\frac{\alpha}{2}-1}\pi^{\frac{\alpha}{4}-\frac{1}{2}}}{2^{\frac{\alpha}{2}-1}\Gamma\left(\frac{\alpha}{2}\right)}$
$\displaystyle+\,\frac{e^{-\frac{y^{2}}{4x}}}{x}\,\sum_{n=0}^{\infty}\tilde{r}_{\alpha}(n)\,\left(n+\frac{\alpha}{4}\right)^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\frac{\pi}{x}\,\left(n+\frac{\alpha}{4}\right)}$
$\displaystyle
I_{\frac{\alpha}{2}-1}\left(\frac{\sqrt{\pi(n+\frac{\alpha}{4})}\,y}{x}\right).$
(2.22)

The previous lemma was necessary in [24] to prove that, for $z$ under the
condition (1.8) and any $m\in\mathbb{N}_{0}$,

$\frac{d^{m}}{d\omega^{m}}\left(1+\psi_{\alpha}\left(e^{2i\omega},z\right)\right)\rightarrow
0,\,\,\,\,\,\text{as \,\,\,$\omega\rightarrow\frac{\pi}{4}^{-}.$}$ (2.23)

This curious transformation formula will now play an essential role in giving
a more precise version of (2.23). This is done in the next lemma, where we
shall estimate uniformly the derivatives of the function
$\psi_{\alpha}\left(ie^{-2iu},z\right)$, $0<u<\frac{\pi}{4}$.

######  Lemma 2.2.

Let $\psi_{\alpha}(x,z)$ be the generalized Jacobi’s $\psi-$function defined
by (1.11) and assume that $z\in\mathbb{R}$ satisfies the condition

$-\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}}\leq
z\leq\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}}.$ (2.24)

Then there are two positive constants $A$ and $C$ (depending only on $\alpha$)
such that, for any $0<u<\frac{\pi}{4}$,

$\left|\frac{d^{k}}{du^{k}}\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right|<C\,\frac{2^{7k}k!}{u^{\frac{\alpha}{2}+k}}e^{-\frac{A}{u}},\,\,\,\,\alpha>2$
(2.25)

and

$\left|\frac{d^{k}}{du^{k}}\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right|<C\,\frac{2^{7k}k!}{u^{\frac{\alpha}{2}+k+2}}e^{-\frac{A}{u}},\,\,\,\,0\leq\alpha\leq
2.$ (2.26)

###### Proof.

Let us fix $0<u_{0}<\frac{\pi}{4}$. We shall prove (2.25) only, as the bound
(2.26) can be similarly deduced. At the end of the proof we just outline the
difference in getting (2.26). The derivative of the function
$\psi_{\alpha}\left(ie^{-2iu},z\right)$ at the point $u_{0}$ is taken by
integrating along a circle with center $u_{0}$ and having radius $\lambda
u_{0}$, $0<\lambda<1$. Throughout this proof, the parameter $\lambda$ will be
arbitrary up to the point where the condition (2.24) enters in the argument.
For any $w\in D_{\lambda u_{0}}(u_{0}):=\text{int}(C_{\lambda u_{0}}(u_{0}))$,
it is simple to check that $\text{Re}\left(ie^{-2iw}\right)>0$, so that
$\psi_{\alpha}\left(ie^{-2iw},z\right)$ is analytic inside the circle
$C_{\lambda u_{0}}(u_{0})$ [[24], Corollary 2.2]. Hence, by Cauchy’s formula,

$\left[\frac{d^{k}}{du^{k}}\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right]_{u=u_{0}}=\frac{k!}{2\pi
i}\,\intop_{C_{\lambda
u_{0}}(u_{0})}\,\frac{1+\psi_{\alpha}\left(ie^{-2iw},z\right)}{(w-u_{0})^{k+1}}\,dw.$
(2.27)

Now, let us bound trivially the integral above: to do it, it will be crucial
to employ the transformation formula (2.22). Indeed, since
$ie^{-2iw}=i+2e^{-iw}\sin(w)$ and, by definition (1.11),

$1+\psi_{\alpha}(ie^{-2iw},z):=1+2^{\frac{\alpha}{2}-1}\,\Gamma\left(\frac{\alpha}{2}\right)\,\left(\sqrt{\pi}\,e^{i\frac{\pi}{4}}e^{-iw}\,z\right)^{1-\frac{\alpha}{2}}\,\sum_{n=1}^{\infty}(-1)^{n}r_{\alpha}(n)\,n^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-2\pi
ne^{-iw}\sin(w)}\,J_{\frac{\alpha}{2}-1}\left(\sqrt{\pi
n}\,e^{i\left(\frac{\pi}{4}-w\right)}\,z\right),$

an application of (2.22) with $x=2e^{-iw}\sin(w)$ and
$y=e^{i\left(\frac{\pi}{4}-w\right)}\,z$ gives

$\displaystyle 1+\psi_{\alpha}(ie^{-2iw},z)$
$\displaystyle=2^{\frac{\alpha}{2}-1}\Gamma\left(\frac{\alpha}{2}\right)\left(\sqrt{\pi}\,e^{i\left(\frac{\pi}{4}-w\right)}z\right)^{1-\frac{\alpha}{2}}$
$\displaystyle\times\,\frac{e^{-\frac{ie^{-iw}z^{2}}{8\sin(w)}}}{2e^{-iw}\sin(w)}$
$\displaystyle\,\sum_{n=0}^{\infty}\,\tilde{r}_{\alpha}(n)e^{-\frac{\pi
i}{2}\left(n+\frac{\alpha}{4}\right)}\,\left(n+\frac{\alpha}{4}\right)^{\frac{1}{2}-\frac{\alpha}{4}}\,e^{-\frac{\pi}{2\tan(w)}\,\left(n+\frac{\alpha}{4}\right)}I_{\frac{\alpha}{2}-1}\left(\frac{\sqrt{\pi(n+\frac{\alpha}{4})\,}e^{\frac{i\pi}{4}}z}{2\sin(w)}\right).$
(2.28)

We shall bound $|1+\psi_{\alpha}\left(ie^{-2iw},z\right)|$ by estimating the
second expression on the right-hand side of (2.28). We will track each factor
independently. Clearly, for any $w\in C_{\lambda u_{0}}(u_{0})$,

$\left|2^{\frac{\alpha}{2}-1}\,\Gamma\left(\frac{\alpha}{2}\right)\,\left(\sqrt{\pi}\,e^{i\frac{\pi}{4}}e^{-iw}z\right)^{1-\frac{\alpha}{2}}\right|\leq
2^{\frac{\alpha}{2}-1}\Gamma\left(\frac{\alpha}{2}\right)\pi^{\frac{1}{2}-\frac{\alpha}{4}}e^{\frac{\pi}{4}\left|1-\frac{\alpha}{2}\right|}|z|^{1-\frac{\alpha}{2}},$

where we have noted that $\lambda u_{0}\leq\frac{\pi}{4}$. Next,

$\displaystyle\left|\frac{e^{-\frac{ie^{-iw}z^{2}}{8\sin(w)}}}{2e^{-iw}\sin(w)}\right|$
$\displaystyle=\frac{e^{-\frac{z^{2}}{8}-\text{Im}(w)}\exp\left(\frac{z^{2}}{8}\text{Im}\left(\frac{1}{\tan(w)}\right)\right)}{2\sqrt{\sin^{2}\left(\text{Re}(w)\right)+\sinh^{2}\left(\text{Im}(w)\right)}}$
$\displaystyle=\frac{e^{-\frac{z^{2}}{8}-\text{Im}(w)}\exp\left(-\frac{z^{2}}{8}\cdot\frac{\sinh(2\text{Im}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}\right)}{2\sqrt{\sin^{2}\left(\text{Re}(w)\right)+\sinh^{2}\left(\text{Im}(w)\right)}}$
$\displaystyle\leq\frac{e^{-\frac{z^{2}}{8}+\lambda
u_{0}}}{2\sin\left(\text{Re}(w)\right)}\,\exp\left(-\frac{z^{2}}{8}\cdot\frac{\sinh(2\text{Im}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}\right),$

where in the last step we just have used the fact that, for any $w\in
C_{\lambda u_{0}}(u_{0})$, $|\text{Im}(w)|\leq\lambda u_{0}$. Moreover, we
have used the explicit expression

$\frac{1}{\tan(w)}=\frac{\sin(2\text{Re}(w))-i\,\sinh(2\text{Im}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}.$
(2.29)

Using now the elementary Jordan inequality $\sin(x)>\frac{2x}{\pi}$,
$0<x<\frac{\pi}{2}$, and the fact that
$(1-\lambda)u_{0}\leq\text{Re}(w)\leq(1+\lambda)u_{0}<\frac{\pi}{2}$, we
obtain

$\displaystyle\left|\frac{e^{-\frac{ie^{-iw}z^{2}}{8\sin(w)}}}{2e^{-iw}\sin(w)}\right|$
$\displaystyle<\frac{\pi e^{-\frac{z^{2}}{8}+\lambda
u_{0}}}{4\text{Re}(w)}\,\exp\left(-\frac{z^{2}}{8}\cdot\frac{\sinh(2\text{Im}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}\right)$
$\displaystyle\leq\frac{\pi e^{-\frac{z^{2}}{8}+\lambda
u_{0}}}{4(1-\lambda)u_{0}}\,\exp\left(-\frac{z^{2}}{8}\cdot\frac{\sinh(2\text{Im}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}\right),\,\,\,\,w\in
C_{\lambda u_{0}}(u_{0}).$ (2.30)

We now address each term of the series on the right-hand side of (2.28). Since
we first aim at proving (2.25), we assume here that $\alpha>2$, so that
$\frac{\alpha}{2}-1>0$. Using the first uniform inequality for $I_{\nu}(z)$,
(2.14), we have that the term involving the Bessel function admits the bound

$\displaystyle\left|I_{\frac{\alpha}{2}-1}\left(\frac{\sqrt{\pi(n+\frac{\alpha}{4})\,}e^{\frac{i\pi}{4}}z}{2\sin(w)}\right)\right|$
$\displaystyle\leq\frac{\pi^{\frac{\alpha}{4}-\frac{1}{2}}\left(n+\frac{\alpha}{4}\right)^{\frac{\alpha}{4}-\frac{1}{2}}|z|^{\frac{\alpha}{2}-1}}{2^{\alpha-2}\Gamma\left(\frac{\alpha}{2}\right)|\sin(w)|^{\frac{\alpha}{2}-1}}\,\exp\left(\frac{\sqrt{\pi\left(n+\frac{\alpha}{4}\right)}\,|z|}{2\,|\sin(w)|}\right)$
$\displaystyle\leq\frac{\pi^{\frac{\alpha}{4}-\frac{1}{2}}\left(n+\frac{\alpha}{4}\right)^{\frac{\alpha}{4}-\frac{1}{2}}|z|^{\frac{\alpha}{2}-1}}{2^{\alpha-2}\Gamma\left(\frac{\alpha}{2}\right)\sin^{\frac{\alpha}{2}-1}\left(\text{Re}(w)\right)}\,\exp\left(\frac{\sqrt{\frac{\pi}{2}\left(n+\frac{\alpha}{4}\right)}\,|z|}{\sqrt{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}}\right)$
$\displaystyle\leq\frac{\pi^{\frac{3\alpha}{4}-\frac{3}{2}}\left(n+\frac{\alpha}{4}\right)^{\frac{\alpha}{4}-\frac{1}{2}}|z|^{\frac{\alpha}{2}-1}}{2^{\frac{3\alpha}{2}-3}\Gamma\left(\frac{\alpha}{2}\right)\left(1-\lambda\right)^{\frac{\alpha}{2}-1}u_{0}^{\frac{\alpha}{2}-1}}\,\exp\left(\frac{\sqrt{\frac{\pi}{2}\left(n+\frac{\alpha}{4}\right)}\,|z|}{\sqrt{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}}\right).$
(2.31)

In the second inequality above we have used the fact that
$|\sin(w)|\geq|\sin\left(\text{Re}(w)\right)|$ and in the third inequality we
have once more invoked Jordan’s inequality $\sin(x)>\frac{2x}{\pi}$,
$0<x<\frac{\pi}{2}$, together with the fact that
$(1-\lambda)u_{0}\leq\text{Re}(w)\leq(1+\lambda)u_{0}<\frac{\pi}{2}$.
Returning to (2.28) and recalling once more (2.29), from (2.30) and (2.31) we
deduce

$\left|1+\psi_{\alpha}(ie^{-2iw},z)\right|<d_{\alpha}\,\frac{\pi^{\frac{\alpha}{2}}e^{-\frac{z^{2}}{8}+\lambda
u_{0}}}{2^{\alpha}\left(1-\lambda\right)^{\frac{\alpha}{2}}u_{0}^{\frac{\alpha}{2}}}\,\sum_{n=0}^{\infty}|\tilde{r}_{\alpha}(n)|\,\exp\left\\{-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\,P_{w}\left(\sqrt{n+\frac{\alpha}{4}}\right)\right\\},$
(2.32)

where $d_{\alpha}$ is some constant depending on $\alpha$ and
$P_{w}\left(X\right)$ is the real-valued polynomial

$P_{w}(X):=X^{2}-\sqrt{\frac{2}{\pi}}\frac{|z|}{\sin(2\text{Re}(w))}\,\sqrt{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\,X+\frac{z^{2}}{4\pi}\,\frac{\sinh(2\text{Im}(w))}{\sin(2\text{Re}(w))}.$
(2.33)

The inequality (2.32) is valid for any $w\in C_{\lambda u_{0}}(u_{0})$. Taking
out the first term of the series on (2.32) and using the fact that
$\tilde{r}_{\alpha}(0):=2^{\alpha}$, we get from (2.32)

$\displaystyle\left|1+\psi_{\alpha}(ie^{-2iw},z)\right|$
$\displaystyle<\frac{d_{\alpha}\,\pi^{\frac{\alpha}{2}}e^{-\frac{z^{2}}{8}+\frac{u_{0}}{2}}}{\left(1-\lambda\right)^{\frac{\alpha}{2}}u_{0}^{\frac{\alpha}{2}}}\exp\left[-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\,P_{w}\left(\frac{\sqrt{\alpha}}{2}\right)\right]$
$\displaystyle\times\sum_{n=0}^{\infty}\left|\frac{\tilde{r}_{\alpha}(n)}{\tilde{r}_{\alpha}(0)}\right|\exp\left[-Q_{w}(n)\right].$
(2.34)

The exponent showing up in the infinite series is explicitly

$Q_{w}(n):=\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\left[P_{w}\left(\sqrt{n+\frac{\alpha}{4}}\right)-P_{w}\left(\frac{\sqrt{\alpha}}{2}\right)\right],$

where $P_{w}(X)$ is the polynomial given by (2.33). We will show below that
there is a choice of $\lambda$ such that, for any $z$ satisfying (2.24),
$Q_{w}(n)>0,\,\,\forall n\in\mathbb{N}$. Since
$|\tilde{r}_{\alpha}(n)|<A_{\alpha}\,n^{\frac{\alpha}{2}}$ (see [[24], p.
38]), the infinite series that appears on the right-hand side of (2.34) is
convergent. The remaining part of our proof will be to find a uniform bound
for it, with a constant depending on $\lambda$ and $\alpha$. We want to find
an upper bound for

$\sum_{n=0}^{\infty}\left|\frac{\tilde{r}_{\alpha}(n)}{\tilde{r}_{\alpha}(0)}\right|\,\exp\left[-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\left\\{n-\sqrt{\frac{2}{\pi}}\,|z|\frac{\sqrt{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}}{\sin(2\text{Re}(w))}\left(\sqrt{n+\frac{\alpha}{4}}-\frac{\sqrt{\alpha}}{2}\right)\right\\}\right].$

Since $(1-\lambda)u_{0}\leq\text{Re}(w)\leq(1+\lambda)u_{0}$, we know from
Jordan’s inequality

$\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))\geq
1-\cos\left(2(1-\lambda)u_{0}\right)=\intop_{0}^{2(1-\lambda)u_{0}}\sin(t)\,dt>\frac{2}{\pi}\,\intop_{0}^{2(1-\lambda)u_{0}}t\,dt=\frac{4(1-\lambda)^{2}}{\pi}u_{0}^{2},$

which implies

$\frac{1}{\sqrt{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}}<\frac{\sqrt{\pi}}{2(1-\lambda)u_{0}}.$
(2.35)

On the other hand, we can find the upper bound

$\displaystyle\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))$
$\displaystyle=\intop_{0}^{2\text{Im}(w)}\sinh(t)\,dt+\intop_{0}^{2\text{Re}(w)}\sin(t)\,dt\leq\intop_{0}^{2\text{Im}(w)}t\,e^{\frac{t^{2}}{6}}dt+\intop_{0}^{2\text{Re}(w)}t\,dt$
$\displaystyle\leq
2\text{Im}(w)^{2}\,e^{\frac{2}{3}\text{Im}(w)^{2}}+2\text{Re}(w)^{2}<4e^{\frac{2\lambda^{2}}{3}u_{0}^{2}}\left(1+\lambda\right)^{2}u_{0}^{2},$
(2.36)

where we have used the fact that
$(1-\lambda)u_{0}\leq\text{Re}(w)\leq(1+\lambda)u_{0}$, $-\lambda
u_{0}\leq\text{Im}(w)\leq\lambda u_{0}$. We have also used the known
inequality

$\sinh(x)\leq x\,e^{\frac{x^{2}}{6}},\,\,\,\,x>0.$ (2.37)

Since $(1-\lambda)u_{0}\leq\text{Re}(w)\leq(1+\lambda)u_{0}<\frac{\pi}{2}$,
another application of Jordan’s inequality gives

$\sin\left(2\text{Re}(w)\right)>\begin{cases}\frac{4}{\pi}\text{Re}(w),&0<\text{Re}(w)<\frac{\pi}{4}\\\
2-\frac{4}{\pi}\text{Re}(w),&\frac{\pi}{4}\leq\text{Re}(w)<\frac{\pi}{2}\end{cases}\,\,\geq\frac{4}{\pi}\left(1-\lambda\right)u_{0}.$
(2.38)

Thus, combining (2.36) with (2.38) and recalling that $0<u_{0}<\frac{\pi}{4}$,
we get the inequality

$\frac{\sin(2\text{Re}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}>\frac{\left(1-\lambda\right)}{\pi
e^{\frac{2\lambda^{2}}{3}u_{0}^{2}}\left(1+\lambda\right)^{2}u_{0}}>\frac{\left(1-\lambda\right)}{\pi
e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}u_{0}}.$ (2.39)

Thus, the infinite series on the right-hand side of (2.34) admits the bound

$\displaystyle\sum_{n=0}^{\infty}\left|\frac{\tilde{r}_{\alpha}(n)}{\tilde{r}_{\alpha}(0)}\right|\,$
$\displaystyle\exp\left[-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\left\\{n-\frac{\sqrt{\frac{2}{\pi}}\,|z|\left(\sqrt{n+\frac{\alpha}{4}}-\frac{\sqrt{\alpha}}{2}\right)}{\sin(2\text{Re}(w))}\sqrt{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\right\\}\right]$
$\displaystyle<\sum_{n=0}^{\infty}\left|\frac{\tilde{r}_{\alpha}(n)}{\tilde{r}_{\alpha}(0)}\right|\,\exp\left[-\frac{1}{u_{0}}\left\\{\frac{n(1-\lambda)}{2e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi\,|z|}{2\sqrt{2}(1-\lambda)}\left(\sqrt{n+\frac{\alpha}{4}}-\frac{\sqrt{\alpha}}{2}\right)\right\\}\right].$

Using the condition (2.24), one sees that the exponent in the previous
expression is bounded by

$\displaystyle\exp\left[-\frac{1}{u_{0}}\left\\{\frac{n(1-\lambda)}{2e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi\,|z|}{2\sqrt{2}(1-\lambda)}\left(\sqrt{n+\frac{\alpha}{4}}-\frac{\sqrt{\alpha}}{2}\right)\right\\}\right]$
$\displaystyle\leq\exp\left[-\frac{1}{u_{0}}\left\\{\frac{n(1-\lambda)}{2e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi^{\frac{3}{2}}\sqrt{\alpha}}{24(1-\lambda)}\left(\sqrt{n+\frac{\alpha}{4}}-\frac{\sqrt{\alpha}}{2}\right)\right\\}\right]$
$\displaystyle\leq\exp\left[-\frac{n}{u_{0}}\left\\{\frac{1-\lambda}{2e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi^{\frac{3}{2}}}{24(1-\lambda)}\right\\}\right],$

where in the last inequality we have used the mean value theorem for the
function $f(x)=\sqrt{x+\frac{\alpha}{4}}$. If we select the value
$\lambda=0.01$, we have that

$\frac{1-\lambda}{2e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi^{\frac{3}{2}}}{24(1-\lambda)}>\frac{1}{4},$

and so, since $0<u_{0}<\frac{\pi}{4}$,

$\displaystyle\sum_{n=0}^{\infty}\left|\frac{\tilde{r}_{\alpha}(n)}{\tilde{r}_{\alpha}(0)}\right|\,\exp\left[-\frac{1}{u_{0}}\left\\{\frac{n(1-\lambda)}{2e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi\,|z|}{2\sqrt{2}(1-\lambda)}\left(\sqrt{n+\frac{\alpha}{4}}-\frac{\sqrt{\alpha}}{2}\right)\right\\}\right]$
$\displaystyle\leq\sum_{n=0}^{\infty}\left|\frac{\tilde{r}_{\alpha}(n)}{\tilde{r}_{\alpha}(0)}\right|\,\exp\left[-\frac{n}{4u_{0}}\right]<\frac{A_{\alpha}}{2^{\alpha}}\sum_{n=0}^{\infty}n^{\frac{\alpha}{2}}\,\exp\left[-\frac{n}{\pi}\right]\leq
M_{\alpha}.$ (2.40)

Combining (2.40) with (2.34) and using the fact that $0<u_{0}<\frac{\pi}{4}$,
the following bound holds (note that we are now selecting $\lambda$ as equal
to $0.01$)

$\left|1+\psi_{\alpha}(ie^{-2iw},z)\right|<\frac{d_{\alpha}M_{\alpha}e^{\frac{\pi}{8}}\,\pi^{\frac{\alpha}{2}}\,e^{-\frac{z^{2}}{8}}}{(2u_{0})^{\frac{\alpha}{2}}}\,\exp\left[-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\,P_{w}\left(\frac{\sqrt{\alpha}}{2}\right)\right],$
(2.41)

which is almost (2.25) with $k=0$. To conclude the proof in this direction, we
just need to bound uniformly the term

$\exp\left[-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\,P_{w}\left(\frac{\sqrt{\alpha}}{2}\right)\right],\,\,\,\,\,\,w\in
C_{\lambda u_{0}}(u_{0})$

under the hypothesis on $z$ (2.24) and the choice $\lambda=0.01$. This can be
done by appealing to (2.37), (2.35) and (2.39), which give

$\displaystyle\exp$
$\displaystyle\left[-\frac{\pi}{2}\frac{\sin\left(2\text{Re}(w)\right)}{\cosh\left(2\text{Im}(w)\right)-\cos\left(2\text{Re}(w)\right)}\,P_{w}\left(\frac{\sqrt{\alpha}}{2}\right)\right]<\exp\left[-\frac{\alpha}{8}\,\frac{\left(1-\lambda\right)}{e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}u_{0}}+\sqrt{\frac{\alpha}{2}}\,\frac{\pi\,|z|}{4(1-\lambda)u_{0}}+\frac{\pi
z^{2}\lambda
e^{\frac{\pi^{2}\lambda^{2}}{24}}}{16(1-\lambda)^{2}u_{0}}\right]$
$\displaystyle\leq\exp\left[-\frac{\alpha}{u_{0}}\left\\{\frac{1-\lambda}{8e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi^{\frac{3}{2}}}{48(1-\lambda)}-\frac{\pi^{2}\lambda
e^{\frac{\pi^{2}\lambda^{2}}{24}}}{1152(1-\lambda)^{2}}\right\\}\right].$
(2.42)

However, for $\lambda=0.01$, a numerical confirmation gives

$\frac{1-\lambda}{8e^{\frac{\pi^{2}\lambda^{2}}{24}}\left(1+\lambda\right)^{2}}-\frac{\pi^{\frac{3}{2}}}{48(1-\lambda)}-\frac{\pi^{2}\lambda
e^{\frac{\pi^{2}\lambda^{2}}{24}}}{1152(1-\lambda)^{2}}>0.004,$

and so, returning to (2.41), we get from the previous inequality that

$\left|1+\psi_{\alpha}(ie^{-2iw},z)\right|<\frac{d_{\alpha}M_{\alpha}e^{\frac{\pi}{8}}\,\pi^{\frac{\alpha}{2}}e^{-\frac{z^{2}}{8}}}{(2u_{0})^{\frac{\alpha}{2}}}\exp\left[-\frac{0.04\,\alpha}{u_{0}}\right]=\frac{C}{u_{0}^{\alpha/2}}\,e^{-\frac{A}{u_{0}}}$
(2.43)

where $C$ and $A:=0.04\,\alpha$ only depend on $\alpha$. This proves (2.25)
for $k=0$. For the remaining cases, let us invoke Cauchy’s integral formula
(2.27) and use (2.43) to get

$\left|\left[\frac{d^{k}}{du^{k}}\psi_{\alpha}\left(ie^{-2iu},z\right)\right]_{u=u_{0}}\right|\leq\frac{k!}{2\pi\left(\lambda
u_{0}\right)^{k+1}}\,\intop_{C_{\lambda
u_{0}}(u_{0})}\,|1+\psi_{\alpha}\left(ie^{-2iw},z\right)|\,|dw|<C\,\frac{k!}{u_{0}^{\frac{\alpha}{2}+k}\lambda^{k}}e^{-\frac{A}{u_{0}}}.$
(2.44)

Finally, under the choice $\lambda=0.01$, we know $\frac{1}{\lambda}<2^{7}$,
and so we derive our estimate (2.25) in its final form.

To get the second formula (2.26), we use the bound (2.15) for $I_{\nu}(z)$,
$-1<\nu<0$, and we invoke the same kind of inequalities as in (2.31) to obtain

$\displaystyle\left|I_{\frac{\alpha}{2}-1}\left(\frac{\sqrt{\pi(n+\frac{\alpha}{4})\,}e^{\frac{i\pi}{4}}z}{2\sin(w)}\right)\right|$
$\displaystyle<\frac{\pi^{\frac{\alpha}{4}-\frac{1}{2}}\left(n+\frac{\alpha}{4}\right)^{\frac{\alpha}{4}-\frac{1}{2}}|z|^{\frac{\alpha}{2}-1}}{2^{\alpha-2}\Gamma\left(\frac{\alpha}{2}+1\right)|\sin(w)|^{\frac{\alpha}{2}-1}}\exp\left(\frac{\sqrt{\frac{\pi}{2}\left(n+\frac{\alpha}{4}\right)}\,|z|}{\sqrt{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}}\right)\left(1+\frac{\pi^{3}(n+\frac{\alpha}{4})|z|^{2}}{64(1-\lambda)^{2}u_{0}^{2}}\right)$
$\displaystyle\leq\mathcal{A}\,\frac{\pi^{\frac{\alpha}{4}-\frac{1}{2}}\left(n+\frac{\alpha}{4}\right)^{\frac{\alpha}{4}+\frac{1}{2}}|z|^{\frac{\alpha}{2}+1}}{2^{\frac{3\alpha}{2}-3}\Gamma\left(\frac{\alpha}{2}+1\right)\left(1-\lambda\right)^{\frac{\alpha}{2}+1}u_{0}^{\frac{\alpha}{2}+1}}\exp\left(\frac{\sqrt{\frac{\pi}{2}\left(n+\frac{\alpha}{4}\right)}\,|z|}{\sqrt{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}}\right),$
(2.45)

for some absolutely large constant $\mathcal{A}$. There is no formal
difference between (2.45) and (2.31), the only exception being in the power of
$1/u_{0}$. Thus, the same argument carries through with an extra factor of
$1/u_{0}^{2}$. This gives (2.26) and finishes the proof. ∎

######  Remark 2.1.

Of course, the condition (1.22) may be improved if we take a more careful
choice of the parameter $\lambda$ in the previous proof. Moreover, when
$\nu>-\frac{1}{2}$, the uniform bounds (2.14) and (2.15) can be replaced by
the better inequality,

$\left|I_{\nu}(z)\right|\leq\left(\frac{|z|}{2}\right)^{\nu}\frac{e^{|\text{Re}(z)|}}{\Gamma(\nu+1)},\,\,\,\,\,\nu>-\frac{1}{2},$
(2.46)

which can be obtained via the integral representation [[19], p. 252, eq.
(10.32.2)]

$I_{\nu}\left(z\right)=\frac{\left(z/2\right)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\frac{1}{2}\right)}\,\intop_{-1}^{1}\left(1-t^{2}\right)^{\nu-\frac{1}{2}}e^{zt}\,dt,\,\,\,\,\,\nu>-\frac{1}{2}.$

Using (2.46) instead of (2.14) in the proof above, one may enlarge the
interval
$\left[-\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}},\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}}\right]$
in the condition (2.24) to
$\left[-\frac{1}{4}\sqrt{\frac{\pi\alpha}{2}},\frac{1}{4}\sqrt{\frac{\pi\alpha}{2}}\right]$
when $\alpha\geq 1$. Therefore, there are some cases where technical
improvements lead to stronger results than those stated in our Theorems 1.1
and 1.2.

We now give another lemma that will be crucial in the proof of our result.

######  Lemma 2.3.

Let $h:\,\mathbb{C}\longmapsto\mathbb{C}$ be an analytic function. If
$z\in\left[-\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}},\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}}\right]$
then, for every $p\in\mathbb{N}_{0}$, the following relation holds

$\frac{d^{2p}}{du^{2p}}\left\\{h(u)\,\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right\\}=-2\sinh\left(\frac{z^{2}}{8}\right)h^{(2p)}(u)+\mathcal{A}_{\alpha,p}(u,z),$
(2.47)

where, for any $0<u<\frac{\pi}{4}$, $\mathcal{A}_{\alpha,p}(u,z)$ satisfies
the inequalities

$|\mathcal{A}_{\alpha,p}(u,z)|<D\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2p}}\left\|h\right\|_{L^{\infty}(C_{1}(0))},\,\,\,\,\alpha>2,$
(2.48)
$|\mathcal{A}_{\alpha,p}(u,z)|<D\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2+2p}}\left\|h\right\|_{L^{\infty}(C_{1}(0))},\,\,\,\,0\leq\alpha\leq
2,$ (2.49)

with $A$ and $D$ depending only on $\alpha$.

###### Proof.

First observe that

$\displaystyle\frac{d^{2p}}{du^{2p}}\left\\{h(u)\,\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right\\}$
$\displaystyle=-2\sinh\left(\frac{z^{2}}{8}\right)h^{(2p)}(u)+e^{\frac{z^{2}}{8}}\,\frac{d^{2p}}{du^{2p}}\,\left\\{h(u)\,\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right\\}$
$\displaystyle:=-2\sinh\left(\frac{z^{2}}{8}\right)h^{(2p)}(u)+\mathcal{A}_{\alpha,p}(u,z).$
(2.50)

By the well-known Cauchy’s estimates for the coefficients of analytic
functions and (2.25), we obtain

$\displaystyle\left|e^{\frac{z^{2}}{8}}\,\frac{d^{2p}}{du^{2p}}\,\left\\{h(u)\,\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right\\}\right|$
$\displaystyle\leq
e^{\frac{\pi\alpha}{576}}\,\sum_{k=0}^{2p}\left(\begin{array}[]{c}2p\\\
k\end{array}\right)\,\left|h^{(2p-k)}(u)\right|\,\left|\frac{d^{k}}{du^{k}}\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right|$
$\displaystyle\leq
e^{\frac{\pi\alpha}{576}}\left\|h\right\|_{L^{\infty}(D(0,1))}\,\sum_{k=0}^{2p}\left(\begin{array}[]{c}2p\\\
k\end{array}\right)(2p-k)!\,\left|\frac{d^{k}}{du^{k}}\left(1+\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right|$
$\displaystyle<Ce^{\frac{\pi\alpha}{576}}\,\left\|h\right\|_{L^{\infty}(D(0,1))}\frac{(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}}}\,\sum_{k=1}^{2p}\frac{2^{7k}}{u^{k}}$
$\displaystyle<C^{\prime}\,\left\|h\right\|_{L^{\infty}(C_{1}(0))}\frac{(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2p}}\,\frac{(128)^{2p+1}-u^{2p+1}}{128-u}$
$\displaystyle<C^{\prime\prime}\,\left\|h\right\|_{L^{\infty}(C_{1}(0))}\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2p}}$

where we have invoked the maximum modulus principle. This completes the proof
of (2.48). The proof of (2.49) is analogous. ∎

To conclude this section, let us recall the useful notation introduced in [8].
From now on, it will be convenient to write the terms of the sequence
$\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ in polar coordinates

$\frac{i\alpha}{2}-2\lambda_{j}:=r_{j}\,e^{i\theta_{j}},\,\,\,0<\theta_{j}<\pi.$
(2.51)

According to our assumption 1, we need to extend this definition to negative
indices: since $\lambda_{-j}=-\lambda_{j}$, then we have that $r_{-j}=r_{j}$
and $\theta_{-j}=\pi-\theta_{j}$.

Using these polar coordinates, we shall introduce a simple lemma which gives
an integral representation for the moments of
$\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)$. Its proof follows
closely the steps given in [[24], pp. 44-45], so we shall omit some details.

######  Lemma 2.4.

Let $-\frac{\pi}{4}<\omega<\frac{\pi}{4}$, $p\in\mathbb{N}_{0}$ and
$\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)$ be the function defined
by (1.23). Then the following integral representation holds

$\displaystyle\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(2\omega
t\right)\,dt$
$\displaystyle=-\frac{8\pi}{2^{2p}}\,\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\left[\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\omega\alpha}{2}\right)\cosh\left(2\omega\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\omega\alpha}{2}\right)\sinh\left(2\omega\lambda_{j}\right)\right]$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}$
$\displaystyle\left(\frac{d^{2p}}{d\omega^{2p}}\left\\{\sum_{j\neq
0}c_{j}e^{\frac{i\omega\alpha}{2}-2\omega\lambda_{j}}\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(e^{2i\omega},z\right)\right)\right\\}\right),$
(2.52)

where $\psi_{\alpha}(x,z)$ is the generalized Jacobi’s $\psi-$function (1.11)
and $(r_{j},\theta_{j})$ are the polar coordinates attached to
$\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ defined by (2.51).

###### Proof.

We start by using the integral representation given at the introduction
(1.10), replacing there $x$ by $e^{2i\omega}$,
$-\frac{\pi}{4}<\omega<\frac{\pi}{4}$: we obtain the formula

$\frac{1}{2\pi}\,\intop_{-\infty}^{\infty}\eta_{\alpha}\left(\frac{\alpha}{4}+it\right)\,_{1}F_{1}\left(\frac{\alpha}{4}+it;\,\frac{\alpha}{2};\,-\frac{z^{2}}{4}\right)\,e^{2\omega
t}\,dt=e^{\frac{i\omega\alpha}{2}}\,\psi_{\alpha}\left(e^{2i\omega},z\right)-e^{-\frac{i\omega\,\alpha}{2}}e^{-z^{2}/4}.$
(2.53)

Using Kummer’s identity [[1], p. 191, eq. (4.1.11)],

${}_{1}F_{1}\left(a;\,c;\,x\right)=e^{x}\,_{1}F_{1}\left(c-a;\,c;\,-x\right),$
(2.54)

an equivalent version of (2.53) reads

$\frac{e^{-z^{2}/8}}{2\pi}\,\intop_{-\infty}^{\infty}\eta_{\alpha}\left(\frac{\alpha}{4}+it\right)\,_{1}F_{1}\left(\frac{\alpha}{4}-it;\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\,e^{2\omega
t}\,dt=e^{\frac{i\omega\alpha}{2}}\,e^{z^{2}/8}\,\psi_{\alpha}\left(e^{2i\omega},z\right)-e^{-\frac{i\omega\,\alpha}{2}}e^{-z^{2}/8}.$
(2.55)

In [[24], p. 43, eq. (3.42)], we have used (2.55) and the estimates (2.12) to
obtain the following formula

$\displaystyle\intop_{-\infty}^{\infty}t^{2p}\,F_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,e^{2\omega
t}\,dt$
$\displaystyle=-\frac{8\pi}{2^{2p}}\,\sum_{j=1}^{\infty}c_{j}\,r_{j}^{2p}e^{-2\omega\lambda_{j}}\,\cos\left(2p\,\theta_{j}+\frac{\omega\alpha}{2}\right)$
$\displaystyle+\frac{4\pi}{2^{2p}}\,\text{Re}$
$\displaystyle\left(e^{z^{2}/8}\,\frac{d^{2p}}{d\omega^{2p}}\left\\{\sum_{j=1}^{\infty}c_{j}e^{\frac{i\omega\alpha}{2}-2\omega\lambda_{j}}\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(e^{2i\omega},z\right)\right)\right\\}\right),$

where $0<\omega<\frac{\pi}{4}$, $z\in\mathbb{C}$, $\psi_{\alpha}(x,z)$ is the
generalized theta function (1.11) and $F_{z,\alpha}(s)$ is the function given
by (cf. (1.9) above)

$F_{z,\alpha}(s):=\sum_{j=1}^{\infty}c_{j}\,\eta_{\alpha}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{\overline{z}^{2}}{4}\right)\right\\}.$

According to assumption 1, our symmetric shifted combination,
$\tilde{F}_{z,\alpha}(s)$, has the same expression as $F_{z,\alpha}(s)$ with
an additional extension to the negative integers (see (1.20) above).
Therefore, assuming also that $z\in\mathbb{R}$ (see assumption 2), we
immediately find that

$\displaystyle\intop_{-\infty}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,e^{2\omega
t}\,dt$ $\displaystyle=-\frac{8\pi}{2^{2p}}\,\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\omega\lambda_{j}}\,\cos\left(2p\,\theta_{j}+\frac{\omega\alpha}{2}\right)$
$\displaystyle+\frac{4\pi}{2^{2p}}e^{z^{2}/8}\,\text{Re}$
$\displaystyle\left(\frac{d^{2p}}{d\omega^{2p}}\left\\{\sum_{j\neq
0}c_{j}e^{\frac{i\omega\alpha}{2}-2\omega\lambda_{j}}\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(e^{2i\omega},z\right)\right)\right\\}\right).$
(2.56)

But it is clear that the first infinite series admits the expression

$\displaystyle\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\omega\lambda_{j}}\,\cos\left(2p\,\theta_{j}+\frac{\omega\alpha}{2}\right)=\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\left[e^{-2\omega\lambda_{j}}\cos\left(2p\,\theta_{j}+\frac{\omega\alpha}{2}\right)+e^{2\omega\lambda_{j}}\cos\left(2p\,\theta_{j}-\frac{\omega\alpha}{2}\right)\right]$
$\displaystyle=2\,\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\left[\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\omega\alpha}{2}\right)\cosh\left(2\omega\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\omega\alpha}{2}\right)\sinh\left(2\omega\lambda_{j}\right)\right],$
(2.57)

because $\lambda_{-j}=-\lambda_{j}$, $r_{-j}=r_{j}$ and
$\theta_{-j}=\pi-\theta_{j}$ by hypothesis. Since
$\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)$ is a real and an even
function of $t$, using the previous expression allows to rewrite (2.56) in the
form

$\displaystyle\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(2\omega
t\right)\,dt$
$\displaystyle=-\frac{8\pi}{2^{2p}}\,\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\left[\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\omega\alpha}{2}\right)\cosh\left(2\omega\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\omega\alpha}{2}\right)\sinh\left(2\omega\lambda_{j}\right)\right]$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}$
$\displaystyle\left(\frac{d^{2p}}{d\omega^{2p}}\left\\{\sum_{j\neq
0}c_{j}e^{\frac{i\omega\alpha}{2}-2\omega\lambda_{j}}\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(e^{2i\omega},z\right)\right)\right\\}\right),$

which completes the proof. ∎

##  3 Proof of Theorem 1.1

###  3.1 Outline of the Proof

At the core of our method is the following lemma, which is an immediate
consequence of a Theorem of Fejér [31, 9, 10].

######  Lemma 3.1.

Let $\left(p_{n}\right)_{n\in\mathbb{N}_{0}}$ be a subsequence of
$\mathbb{N}_{0}$ with $p_{0}:=0$. Then the number of variations of sign in the
interval $[0,a]$ of a real continuous function $f(x)$ is not less than the
number of variations of sign of the following sequence

$\mathcal{F}_{p_{n}}(a):=\begin{cases}f(0)&n=0\\\
\intop_{0}^{a}f(t)\,t^{p_{n}-1}dt&n\geq 1.\end{cases}$

Our proof of Theorem 1.1 is essentially divided in four parts: in the first of
these, we employ lemmas 2.3 and 2.4 to rewrite (2.52) in a form which is
useful to apply the zero counting method suggested by the previous lemma. By
the conditions of our Theorem 1.1, the sequence
$\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ attains its bounds and is made of
distinct elements. Hence, we know that exists some $M\in\mathbb{N}$ such that

$|\lambda_{M}|=\max_{j\in\mathbb{N}}\\{|\lambda_{j}|\\},\,\,\,\,\,\lambda_{j}\neq\lambda_{M}\,\,\,\text{for}\,\,\,j\neq
M.$ (3.1)

This implies that $r_{j}<r_{M}$ for $j\neq M$ by definition of the polar
coordinates (2.51). During the proof of our result, we will assume without any
loss of generality that $\lambda_{M}<0$.333There is no loss of generality
because we can interchange $\lambda_{j}$ with $-\lambda_{j}$ in the function
$\tilde{F}_{z,\alpha}\left(s\right)$. According to the representation (2.51),
this assumption implies that $\theta_{M}\in\left(0,\frac{\pi}{2}\right)$.

Using the existence of $\lambda_{M}$ given in (3.1), the first section of our
proof is devoted to show that, for $0<u<\frac{\pi}{4},$

$\displaystyle\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle=-\frac{8\pi
c_{M}r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\mathcal{G}_{p}\left(\theta_{M}\right)$
$\displaystyle\times\left\\{1+\frac{\mathcal{B}_{\alpha,p}(u)}{2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)}+\tilde{E}\left(X,z\right)+\tilde{H}\left(X,z\right)\right\\}$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}\left[\mathcal{A}_{\alpha,p}(u,z)\right],$
(3.2)

where $\mathcal{A}_{\alpha,p}(u,z)$ and the terms inside the braces will be
specified later. Moreover, $\theta_{M}$ is the angular coordinate of
$\frac{i\alpha}{2}-2\lambda_{M}:=r_{M}e^{i\theta_{M}}$ in the convention
(2.51) and $(\mathcal{G}_{p}(\theta_{M}))_{p\in\mathbb{N}}$ is the sequence
given by

$\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}:=\cos(2p\theta_{M})\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi\lambda_{M}}{2}\right)+\sin(2p\theta_{M})\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi\lambda_{M}}{2}\right).$
(3.3)

In the second part we construct a sequence of integers
$\left(q_{n}\right)_{n\in\mathbb{N}}$ such that, for every $n\in\mathbb{N}$,
$\mathcal{G}_{q_{n}}(\theta_{M})$ and $\mathcal{G}_{q_{n+1}}(\theta_{M})$ have
always different signs. In the third part of our proof, we prove that the
terms

$\mathcal{A}_{\alpha,p}(u,z),\,\frac{\mathcal{B}_{\alpha,p}(u)}{2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)},\,\tilde{E}\left(X,z\right),\,\tilde{H}\left(X,z\right)$

are very small if we pick $u$ small enough and take $p$ as a very large
integer. Finally, in the fourth part of our proof we combine our collected
data with Fejér’s lemma 3.1 to reach the desired conclusion.

We should remark that, throughout this proof, we will only work with the
assumption that $\alpha>2$, so that we will apply bounds (2.25) and (2.48)
respectively. Of course, in the range $0\leq\alpha\leq 2$ the proof is
entirely analogous but the auxiliary computations are slightly different.

###  3.2 A suitable integral representation

Let $0<u<\frac{\pi}{4}$ and take $\omega=\frac{\pi}{4}-u$ on the integral
representation (2.52). If $\mathcal{I}_{2p}(u)$ denotes the integral

$\mathcal{I}_{2p}(u):=\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt,\,\,\,\,0<u<\frac{\pi}{4},$

then, according to (2.52), $\mathcal{I}_{2p}(u)$ can be explicitly given by

$\displaystyle-\frac{8\pi}{2^{2p}}$
$\displaystyle\,\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\left[\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\alpha}{2}\left(\frac{\pi}{4}-u\right)\right)\cosh\left(2\lambda_{j}\left(\frac{\pi}{4}-u\right)\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\alpha}{2}\left(\frac{\pi}{4}-u\right)\right)\sinh\left(2\lambda_{j}\left(\frac{\pi}{4}-u\right)\right)\right]$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}\left(\frac{d^{2p}}{du^{2p}}\left\\{e^{\frac{i\alpha}{2}\left(\frac{\pi}{4}-u\right)}\sum_{j\neq
0}c_{j}e^{-2\lambda_{j}\,\left(\frac{\pi}{4}-u\right)}\,\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\right\\}\right)=\mathcal{I}_{2p}(u).$
(3.4)

We shall rewrite (3.4) by appealing to Lemma 2.3. Consider the following
function

$h_{\alpha}(u)=e^{\frac{i\alpha}{2}\left(\frac{\pi}{4}-u\right)}\sum_{j\neq
0}c_{j}e^{-2\lambda_{j}\,\left(\frac{\pi}{4}-u\right)}.$ (3.5)

Since $\sum|c_{j}|<\infty$, it is clear that $h_{\alpha}(u)$ is an analytic
function of $u$. Therefore, by (2.50) and the notation (2.51),

$\displaystyle\frac{d^{2p}}{du^{2p}}\left\\{\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\,e^{\frac{i\alpha}{2}\left(\frac{\pi}{4}-u\right)}\sum_{j\neq
0}c_{j}e^{-2\lambda_{j}\,\left(\frac{\pi}{4}-u\right)}\right\\}$
$\displaystyle=-2\sinh\left(\frac{z^{2}}{8}\right)h_{\alpha}^{(2p)}(u)+\mathcal{A}_{\alpha,p}(u,z)$
$\displaystyle=-2\sinh\left(\frac{z^{2}}{8}\right)\,e^{\frac{i\alpha}{2}\left(\frac{\pi}{4}-u\right)}\,\sum_{j\neq
0}c_{j}r_{j}^{2p}e^{2ip\theta_{j}}\,$ $\displaystyle
e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}+\mathcal{A}_{\alpha,p}(u,z).$
(3.6)

We will now find a suitable bound for $\mathcal{A}_{\alpha,p}(u,z)$, which
will invoke Lemma 2.3. Since $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$ is
bounded by hypothesis, there exists some $\mu$ such that $|\lambda_{j}|<\mu$
for every $j\in\mathbb{N}$: hence

$\left\|h_{\alpha}\right\|_{L^{\infty}(C_{0}(1))}\leq
e^{\frac{\alpha}{2}}\sum_{j\neq
0}|c_{j}|e^{\left(2-\frac{\pi}{2}\right)\lambda_{j}}<e^{\left(4-\pi+\alpha\right)\mu}\,\sum_{j\neq
0}|c_{j}|\leq\mathcal{M},$ (3.7)

where $\mathcal{M}$ only depends on the sequence the value of the convergent
series $\sum_{j\neq 0}|c_{j}|:=2\sum_{j=1}^{\infty}|c_{j}|$. Recalling (2.48),
we arrive at the following bound

$|\mathcal{A}_{\alpha,p}(u,z)|<D\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2p}}\left\|h\right\|_{L^{\infty}(C_{1}(0))}\leq\mathscr{D}\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2p}},$
(3.8)

for some $\mathscr{D}$ depending only on the sequence
$\left(c_{j}\right)_{j\in\mathbb{N}}$ and $\alpha$. Returning to the second
term of (3.4) and using (3.6), we find that

$\displaystyle
e^{z^{2}/8}\text{Re}\left(\,\frac{d^{2p}}{du^{2p}}\left\\{\left(e^{-z^{2}/8}+e^{z^{2}/8}\,\psi_{\alpha}\left(ie^{-2iu},z\right)\right)\,e^{\frac{i\alpha}{2}\left(\frac{\pi}{4}-u\right)}\sum_{j\neq
0}c_{j}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\right\\}\right)$
$\displaystyle=e^{z^{2}/8}\,\text{Re}\left[\mathcal{A}_{\alpha,p}(u,z)\right]-2e^{\frac{z^{2}}{8}}\sinh\left(\frac{z^{2}}{8}\right)\,\sum_{j\neq
0}c_{j}r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\,\cos\left(2p\theta_{j}+\frac{\pi\alpha}{8}-\frac{\alpha}{2}u\right).$
(3.9)

We can rewrite the infinite series on the previous expression by repeating the
same computations leading to (2.57): we see that

$\displaystyle
2\,\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\left\\{\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\alpha}{2}\left(\frac{\pi}{4}-u\right)\right)\cosh\left(2\left(\frac{\pi}{4}-u\right)\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\alpha}{2}\left(\frac{\pi}{4}-u\right)\right)\sinh\left(2\left(\frac{\pi}{4}-u\right)\lambda_{j}\right)\right\\}$
$\displaystyle=\sum_{j\neq
0}c_{j}r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\,\cos\left(2p\theta_{j}+\frac{\pi\alpha}{8}-\frac{\alpha}{2}u\right),$
(3.10)

and so, after combining (3.10) with (3.4), we derive the representation

$\displaystyle\mathcal{I}_{2p}(u)=\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}\left[\mathcal{A}_{\alpha,p}(u,z)\right]-\frac{8\pi}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)$
$\displaystyle\times\sum_{j=1}^{\infty}c_{j}\,r_{j}^{2p}\left\\{\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\alpha}{2}\left(\frac{\pi}{4}-u\right)\right)\cosh\left(2\left(\frac{\pi}{4}-u\right)\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\alpha}{2}\left(\frac{\pi}{4}-u\right)\right)\sinh\left(2\left(\frac{\pi}{4}-u\right)\lambda_{j}\right)\right\\}.$
(3.11)

Of course, by (3.9) and (3.4), an equivalent way of writing (3.11) is
returning to the sum over $j\in\mathbb{Z}\setminus\\{0\\}$ and get

$\displaystyle\mathcal{I}_{2p}(u)$
$\displaystyle=-\frac{4\pi}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\,\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\,\cos\left(2p\,\theta_{j}+\frac{\pi\alpha}{8}-\frac{\alpha}{2}u\right)$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}\left[\mathcal{A}_{\alpha,p}(u,z)\right].$
(3.12)

We will now analyze the infinite series on the first term on the right side of
(3.11) and approximate it by its value for $u=0$. For this analysis it will be
easier to manipulate the more compact expression (3.12). By the mean value
theorem and the existence of an element $\lambda_{M}$ satisfying (3.1), we
know that

$\left|\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\,\left\\{\cos\left(2p\,\theta_{j}+\frac{\pi\alpha}{8}-\frac{\alpha}{2}u\right)-\cos\left(2p\,\theta_{j}+\frac{\pi\alpha}{8}\right)\right\\}\right|\leq\frac{\alpha}{2}e^{\frac{\pi}{2}|\lambda_{M}|}r_{M}^{2p}\sum_{j\neq
0}|c_{j}|\,u\leq C_{\alpha}\,r_{M}^{2p}\,u,$ (3.13)

where, by the definition (2.51),
$\frac{i\alpha}{2}-2\lambda_{M}=r_{M}e^{i\theta_{M}}$. Thus, we can write the
infinite series in (3.12) in the approximate form

$\displaystyle\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}$
$\displaystyle\,\cos\left(2p\,\theta_{j}+\frac{\pi\alpha}{8}-\frac{\alpha}{2}u\right)=\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\cos\left(2p\,\theta_{j}+\frac{\pi\alpha}{8}\right)+\mathcal{B}_{\alpha,p}(u)$
$\displaystyle=2\,\sum_{j=1}^{\infty}c_{j}\,r_{j}^{2p}$
$\displaystyle\left\\{\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{j}\right)\right\\}+\mathcal{B}_{\alpha,p}(u),$
(3.14)

where, according to (3.13), $\mathcal{B}_{\alpha,p}(u)$ satisfies the estimate

$|\mathcal{B}_{\alpha,p}(u)|\leq C_{\alpha}\,r_{M}^{2p}\,u.$ (3.15)

Assume that $p\in\mathbb{N}$ is such that

$\mathcal{G}_{p}(\theta_{M}):=\cos\left(2p\theta_{M}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{M}\right)+\sin\left(2p\theta_{M}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{M}\right)\neq
0.$ (3.16)

Then, for every $p$ satisfying (3.16), we can write (3.14) as

$\displaystyle\sum_{j\neq
0}c_{j}\,r_{j}^{2p}e^{-2\left(\frac{\pi}{4}-u\right)\lambda_{j}}\,\cos\left(2p\,\theta_{j}+\frac{\pi\alpha}{8}-\frac{\alpha}{2}u\right)$
$\displaystyle=2\,\sum_{j=1}^{\infty}c_{j}r_{j}^{2p}\,\mathcal{G}_{p}\left(\theta_{j}\right)+\mathcal{B}_{\alpha,p}(u)$
$\displaystyle=2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)$
$\displaystyle\left\\{1+\frac{\mathcal{B}_{\alpha,p}(u)}{2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)}+\tilde{E}\left(X,z\right)+\tilde{H}\left(X,z\right)\right\\},$
(3.17)

where

$\tilde{E}(X,z):=\sum_{j\neq M,\,j\leq
X}\frac{c_{j}}{c_{M}}\,\left(\frac{r_{j}}{r_{M}}\right)^{2p}\,\frac{\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{j}\right)}{\cos\left(2p\theta_{M}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{M}\right)+\sin\left(2p\theta_{M}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{M}\right)}$
(3.18)

and

$\tilde{H}\left(X,z\right):=\sum_{j\neq
M,\,j>X}\frac{c_{j}}{c_{M}}\,\left(\frac{r_{j}}{r_{M}}\right)^{2p}\,\frac{\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{j}\right)}{\cos\left(2p\theta_{M}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{M}\right)+\sin\left(2p\theta_{M}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{M}\right)},$
(3.19)

for some large parameter $X$ that will be chosen later. Using (3.12), we
arrive at the expression

$\displaystyle\mathcal{I}_{2p}(u)$ $\displaystyle=-\frac{8\pi
c_{M}r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\mathcal{G}_{p}\left(\theta_{M}\right)\left\\{1+\frac{\mathcal{B}_{\alpha,p}(u)}{2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)}+\tilde{E}\left(X,z\right)+\tilde{H}\left(X,z\right)\right\\}$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}\left[\mathcal{A}_{\alpha,p}(u,z)\right],$
(3.20)

which is exactly the one claimed at the introduction of our proof, (3.2).

In the remaining parts of our proof, the strategy will be to see that
$\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$ is the dominant
term in (3.17) and in (3.20). This will be concluded after showing that
$\mathcal{A}_{\alpha,p}(u,z)$, $\mathcal{B}_{\alpha,p}(u)$, $\tilde{E}(X,z)$
and $\tilde{H}(X,z)$ are very small for infinitely many values of $p$ and for
sufficiently small $u$. Since (3.3) will change its sign infinitely often for
a suitable sequence of integers $\left(q_{n}\right)_{n\in\mathbb{N}}$, our
proof will be concluded after this observation.

###  3.3 Studying the sign changes of
$\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$

We will now show that we can always construct a sequence
$(q_{n})_{n\in\mathbb{N}}\subset\mathbb{N}$ for which
$\mathcal{G}_{q_{n}}(\theta_{M})$ and $\mathcal{G}_{q_{n+1}}(\theta_{M})$
always have a different sign. To that end, we divide our construction in two
cases: $\frac{\theta_{M}}{\pi}\notin\mathbb{Q}$ or
$\frac{\theta_{M}}{\pi}\in\mathbb{Q}$.

####  $1^{\text{st}}$ case: $\theta_{M}/\pi\notin\mathbb{Q}$

Since we are assuming that $\lambda_{M}<0$ (so that
$\theta_{M}\in(0,\frac{\pi}{2})$), we may write the sequence (3.3) as follows

$\left(\mathcal{G}_{p}\left(\theta_{M}\right)\right)_{p\in\mathbb{N}}=\cos(2p\theta_{M})\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi|\lambda_{M}|}{2}\right)-\sin(2p\theta_{M})\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi|\lambda_{M}|}{2}\right).$

We will study the sign changes of the previous sequence through the study of
the function

$G(\phi):=\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi|\lambda_{M}|}{2}\right)\,\cos(\phi)-\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi|\lambda_{M}|}{2}\right)\,\sin(\phi).$

In order to study $G(\phi)$, let us assume that $8k<\alpha<8k+4$, for some
$k\in\mathbb{N}_{0}$. The remaining cases $8k+4<\alpha<8k+8$ and $\alpha\equiv
0\mod 4$ are analogous and will be sketched at the end of the argument. Under
the assumption $8k<\alpha<8k+4$, we know that
$\cos\left(\frac{\pi\alpha}{8}\right)$ and
$\sin\left(\frac{\pi\alpha}{8}\right)$ must have the same sign and, without
loss of generality, we assume it to be positive.

Let us define the sets

$\mathcal{A}^{+}:=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)-\pi,\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)\right)$
(3.21)

and

$\mathcal{A}^{-}:=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right),\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\pi\right).$
(3.22)

Clearly, if $\phi\in\mathcal{A}^{+}$ then $G(\phi)>0$, while $G(\phi)<0$ when
$\phi\in\mathcal{A}^{-}$.We now consider the subintervals of $\mathcal{A}^{+}$
and $\mathcal{A}^{-}$ defined by

$\mathcal{I}^{+}:=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)-\frac{\pi}{2}-\theta_{M};\,\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)-\frac{\pi}{2}+\theta_{M}\right),$
(3.23)
$\mathcal{I}^{-}:=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\frac{\pi}{2}-\theta_{M};\,\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\frac{\pi}{2}+\theta_{M}\right).$
(3.24)

Then $\mathcal{I}^{+}$ (resp. $\mathcal{I}^{-}$) is an arc in the center of
$\mathcal{A}^{+}$ (resp. $\mathcal{A}^{-}$) with length $2\theta_{M}$. We can
therefore write the partition

$\mathcal{A}^{+}=\mathcal{A}_{1}^{+}\cup\mathcal{I}^{+}\cup\mathcal{A}_{2}^{+},$
(3.25)

where

$\mathcal{A}_{1}^{+}=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)-\pi,\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)-\frac{\pi}{2}-\theta_{M}\right),$

while

$\mathcal{A}_{2}^{+}=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)-\frac{\pi}{2}+\theta_{M},\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)\right).$

Analogously, we can find the partition of $\mathcal{A}^{-}$,

$\mathcal{A}^{-}=\mathcal{A}_{1}^{-}\cup\mathcal{I}^{-}\cup\mathcal{A}_{2}^{-},$

where

$\mathcal{A}_{1}^{-}=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right),\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\frac{\pi}{2}-\theta_{M}\right)$

and

$\mathcal{A}_{2}^{-}=\left(\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\frac{\pi}{2}+\theta_{M},\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\pi\right).$

Figure 1: The partition of the unit circle into $\mathcal{A}^{+}$ and
$\mathcal{A}^{-}$ and the representations of the subsets
$\mathcal{A}_{1}^{\pm}$, $\mathcal{A}_{2}^{\pm}$ and $\mathcal{I}^{\pm}$. The
line dividing the unit circle is defined by the equation
$Y=\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)\,X$.

Since $\theta_{M}/\pi\notin\mathbb{Q}$, by Kronecker’s lemma the set of points
$\left\\{n\theta_{M}/\pi\right\\}_{n\in\mathbb{N}}$ is dense in $(0,1)$. Thus,
there exist two infinite sequences $\left(q_{n}^{+}\right)_{n\in\mathbb{N}}$
and $\left(q_{n}^{-}\right)_{n\in\mathbb{N}}$ such that

$\left(q_{n}^{+}\right)_{n\in\mathbb{N}}:=\left\\{q\in\mathbb{N}\,:\,2q\theta_{M}\in\mathcal{I}^{+}\right\\},\,\,\,\,\left(q_{n}^{-}\right)_{n\in\mathbb{N}}:=\left\\{q\in\mathbb{N}\,:\,2q\theta_{M}\in\mathcal{I}^{-}\right\\}.$
(3.26)

The main claim of this section establishes that
$\left(q_{n}^{+}\right)_{n\in\mathbb{N}}$ and
$\left(q_{n}^{-}\right)_{n\in\mathbb{N}}$ interlace.

######  Claim 3.1.

For every $n\in\mathbb{N}$, we have that

$q_{n}^{+}<q_{n}^{-}<q_{n+1}^{+}<q_{n+1}^{-}<q_{n+2}^{+}<\ldots.$

###### Proof.

First, let us observe that, if $a\in\left(q_{n}^{+}\right)_{n\in\mathbb{N}}$
then $a+1\notin\left(q_{n}^{+}\right)_{n\in\mathbb{N}}$, because
$2(a+1)\theta_{M}=2a\theta_{M}+2\theta_{M}$ and the length of
$\mathcal{I}^{+}$ is, by construction, equal to $2\theta_{M}$. Therefore,
either $2(a+1)\theta_{M}\in\mathcal{I}^{-}$ (i.e.,
$a+1\in\left(q_{n}^{-}\right)_{n\in\mathbb{N}}$) or
$2(a+1)\theta_{M}\in\mathcal{A}_{2}^{+}\cup\mathcal{A}_{1}^{-}$. This actually
happens because $2\theta_{M}<\pi$ and, since $2a\theta_{M}\in\mathcal{I}^{+}$
by hypothesis, then

$2a\theta_{M}+2\theta_{M}<\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)+\frac{\pi}{2}+\theta_{M},$

which assures that $2(a+1)\theta_{M}\notin\mathcal{A}_{2}^{-}$. Since
$a\in(q_{n}^{+})_{n\in\mathbb{N}}$ then $a=q_{m_{0}}^{+}$ for some $m_{0}$. If
$a+1\in(q_{n}^{-})_{n\in\mathbb{N}}$, then we have that
$q_{m_{0}}^{+}<q_{m_{0}}^{-}$. On the other hand, if
$2(a+1)\theta_{M}\in\mathcal{A}_{2}^{+}\cup\mathcal{A}_{1}^{-}$ then there is
some $j\geq 2$ such that $2(a+j)\theta_{M}\in\mathcal{I}^{-}$ because the
length of $\mathcal{I}^{-}$ is equal to $2\theta_{M}$. Therefore, the
translation $a+j$, $j\in\mathbb{N}_{0}$, cannot go from $q_{m_{0}}^{+}$ to
$q_{m_{0}+1}^{+}$ without passing first by an element of the sequence
$\left(q_{n}^{-}\right)_{n\in\mathbb{N}}$. This completes the proof of the
claim. ∎

The interlacing property of $q_{n}^{+}$ and $q_{n}^{-}$ shows that we can
construct a new sequence

$\left(q_{n}\right)_{n\in\mathbb{N}}:=\left(q_{n}^{+}\right)_{n\in\mathbb{N}}\cup\left(q_{n}^{-}\right)_{n\in\mathbb{N}}$

in such a way that $q_{2n-1}:=q_{n}^{+}$ and $q_{2n}:=q_{n}^{-}$. Moreover, we
have that $\mathcal{G}_{q_{m}}\left(\theta_{M}\right)$ and
$\mathcal{G}_{q_{m+1}}\left(\theta_{M}\right)$ have opposite signs.

Finally, we remark that the cases $8k+4<\alpha<8k+8$ and $\alpha\equiv 0\mod
4$ are analogous. In the first case, $\cos\left(\frac{\pi\alpha}{8}\right)$
and $\sin\left(\frac{\pi\alpha}{8}\right)$ have opposite signs. However, a
similar strategy as the one given above works and the only modification is
replacing $\mathcal{A}_{1}^{+},\mathcal{I}^{+}\text{ and }\mathcal{A}_{2}^{+}$
respectively by $\mathcal{A}_{1}^{-},\mathcal{I}^{-}\text{ and
}\mathcal{A}_{2}^{-}$.

On the other hand, if $\alpha=8k$ or $\alpha=8k+4$ for some
$k\in\mathbb{N}_{0}$, then

$\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}=(-1)^{k}\,\cosh\left(\frac{\pi|\lambda_{M}|}{2}\right)\,\cos\left(2p\theta_{M}\right),\,\,\,\text{if
\,}\alpha=8k$ (3.27)

and

$\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}=(-1)^{k-1}\,\sinh\left(\frac{\pi|\lambda_{M}|}{2}\right)\,\sin\left(2p\theta_{M}\right),\,\,\,\text{if
}\,\alpha=8k+4.$ (3.28)

In any of these cases a similar construction of the sets
$\mathcal{A}_{1}^{\pm},$ $\mathcal{A}_{2}^{\pm}$ and $\mathcal{I}^{\pm}$ can
be made.

####  $2^{\text{nd}}$ case: $\theta_{M}/\pi\in\mathbb{Q}$

The conclusions given above in the case $\theta_{M}/\pi\in\mathbb{Q}$ are
analogous but instead of arguing via Kronecker’s lemma, the sequences (3.26)
can be explicitly constructed in this case. Since
$|\mathcal{I}^{+}|=|\mathcal{I}^{-}|=2\theta_{M}$, we know that there exist
$q_{1}$ and $q_{2}>q_{1}$ such that $2q_{1}\theta_{M}\in\mathcal{I}^{+}$ and
$2q_{2}\theta_{M}\in\mathcal{I}^{-}$. Since $\theta_{M}=\frac{a}{b}\pi$,
$\left(a,b\right)=1$, then $2q_{1}\theta_{M}\equiv
2\left(q_{1}+nb\right)\theta_{M}\mod 2\pi$ and $2q_{2}\theta_{M}\equiv
2\left(q_{2}+nb\right)\theta_{M}\mod 2\pi$. Hence, if we construct the
sequence $\left(q_{n}\right)_{n\in\mathbb{N}}$ in such a way that444Note that
the interlacing property established by Claim 3.1 is obviously valid in this
case.

$q_{2n-1}:=q_{1}+nb,\,\,\,\,q_{2n}:=q_{2}+nb,$ (3.29)

then
$\mathcal{G}_{q_{n}}(\theta_{M})\cdot\mathcal{G}_{q_{n+1}}(\theta_{M})<0,$
i.e., $\mathcal{G}_{q_{n}}\left(\theta_{M}\right)$ and
$\mathcal{G}_{q_{n+1}}\left(\theta_{M}\right)$ have opposite signs.

###  3.4 The dominance of
$\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$.

Having constructed a sequence $\left(q_{n}\right)_{n\in\mathbb{N}}$ for which
$\left(\mathcal{G}_{q_{n}}(\theta_{M})\right)_{n\in\mathbb{N}}$ is always
alternating its sign, we are now ready to proceed with the proof. Our next
claim establishes that the sequence $\mathcal{G}_{q_{n}}(\theta_{M})$
dominates, in some sense, the second term (involving
$\mathcal{A}_{\alpha,p}(u,z)$) appearing in (3.20).

######  Claim 3.2.

There exists a sufficiently large $\eta_{0}$ such that, for any $z$ satisfying
(2.24), $\mathcal{A}_{\alpha,p}(\cdot,z)$ satisfies

$e^{z^{2}/8}\,\left|\text{Re}\left[\mathcal{A}_{\alpha,p}\left(\frac{1}{\eta_{0}\,p\log(p)},z\right)\right]\right|<|c_{M}|\,r_{M}^{2p}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\left|\mathcal{G}_{p}\left(\theta_{M}\right)\right|$
(3.30)

for any $p\in\left(q_{n}\right)_{n\in\mathbb{N}}$.

###### Proof.

By (3.8), we have that

$|\mathcal{A}_{\alpha,p}(u,z)|\leq\mathscr{D}\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{\frac{\alpha}{2}+2p}},\,\,\,\,0<u<\frac{\pi}{4},$

so that the inequality (3.30) holds for any
$p\in\left(q_{n}\right)_{n\in\mathbb{N}}$ if

$\frac{\mathscr{D}2^{14p}(2p)!\,e^{z^{2}/8}}{|c_{M}|r_{M}^{2p}\,u^{\frac{\alpha}{2}+2p}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\left|\mathcal{G}_{p}\left(\theta_{M}\right)\right|}<e^{A/u}$
(3.31)

holds for any $p\in(q_{n})_{n\in\mathbb{N}}$. By construction of the sets
$\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ (see (3.23) and (3.24) above), if
$p\in(q_{n})_{n\in\mathbb{N}}$ then

$\left|\mathcal{G}_{p}\left(\theta_{M}\right)\right|=\left|\cos\left(2p\theta_{M}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{M}\right)+\sin\left(2p\theta_{M}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{M}\right)\right|\geq\epsilon_{0},$
(3.32)

for some $\epsilon_{0}>0$ (only depending on $\theta_{M}$). Therefore, since
$e^{z^{2}/8}<e^{\frac{\pi\alpha}{576}}$ (by our condition (1.22)), (3.31) is
plainly satisfied if, for any $p\in(q_{n})_{n\in\mathbb{N}}$,

$\frac{\mathscr{D}\exp\left(\frac{\pi\alpha}{576}\right)}{\epsilon_{0}|c_{M}|}\,\frac{2^{14p}(2p)!}{r_{M}^{2p}u^{\frac{\alpha}{2}+2p}}<e^{A/u}.$
(3.33)

Letting $u=\frac{1}{\eta_{0}\,p\log(p)}$ and taking the logarithm on both
sides of (3.33), we see that (3.33) holds if

$\frac{1}{Ap\log(p)}\log\left(\frac{\mathscr{D}\exp\left(\frac{\pi\alpha}{576}\right)}{\epsilon_{0}|c_{M}|}\right)+\frac{2\,\log\left(128/r_{M}\right)}{A\log(p)}+\frac{\log(2p)!}{Ap\log(p)}+\frac{1}{A\log(p)}\left(\log(\eta)+\log(p)+\log(\log(p))\right)\left(2+\frac{\alpha}{2p}\right)<\eta_{0}.$
(3.34)

From Stirling’s formula, we know that $\log(2p)!=O\left(p\log(p)\right)$, from
which it follows that (3.34) holds for every $p\in(q_{n})_{n\in\mathbb{N}}$ if
$\eta_{0}$ is a sufficiently large number. This completes the proof of (3.30).
∎

We have proved that the second term of (3.20) is dominated by the term
involving $\mathcal{G}_{p}\left(\theta_{M}\right)$ once $p$ belongs to a
suitable sequence of integers. Similar to what we have done in [[24], p. 47],
a bound for the functions $\tilde{E}(X,z)$ and $\tilde{H}(X,z)$ also holds.

######  Claim 3.3.

There exists a sufficiently large $N_{0}$ such that, for any
$p\in\left(q_{n}\right)_{n\geq N_{0}}$ and $z$ satisfying (1.22), the
following inequalities hold

$|\tilde{E}(X,z)|<\frac{1}{6},\,\,\,|\tilde{H}\left(X,z\right)|<\frac{1}{6}$
(3.35)

and, for any $\eta>0$,

$\frac{\left|\mathcal{B}_{\alpha,p}\left(\frac{1}{\eta
p\log(p)}\right)\right|}{2\,|c_{M}|r_{M}^{2p}\left|\mathcal{G}_{p}\left(\theta_{M}\right)\right|}<\frac{1}{6},$
(3.36)

with $\mathcal{B}_{\alpha,p}(u)$ is defined by (3.14).

###### Proof.

Since $p\in(q_{n})_{n\geq N_{0}}$, then (3.32) holds for some
$\epsilon_{0}\geq 0$. Thus, since $(c_{j})_{j\in\mathbb{N}}\in\ell^{1}$, we
can choose a sufficiently large $X\geq X_{0}$ such that

$\displaystyle|\tilde{H}(X,z)|$
$\displaystyle\leq\frac{1}{\epsilon_{0}}\sum_{j\neq
M,\,j>X}\frac{|c_{j}|}{|c_{M}|}\,\left(\frac{r_{j}}{r_{M}}\right)^{2p}\,\left|\cos\left(2p\theta_{j}\right)\,\cos\left(\frac{\pi\alpha}{8}\right)\cosh\left(\frac{\pi}{2}\lambda_{j}\right)+\sin\left(2p\theta_{j}\right)\,\sin\left(\frac{\pi\alpha}{8}\right)\sinh\left(\frac{\pi}{2}\lambda_{j}\right)\right|$
$\displaystyle\leq\frac{2e^{\frac{\pi}{2}\lambda_{M}}}{\epsilon_{0}|c_{M}|}\,\sum_{j\neq
M,j>X}\,|c_{j}|<\frac{1}{6}.$

On the other hand, we may bound $\tilde{E}(X,z)$ as follows

$\left|\tilde{E}(X,z)\right|\leq\frac{2e^{\frac{\pi}{2}\lambda_{M}}}{\epsilon_{0}|c_{M}|}\mu_{X}^{2p}\sum_{j\neq
M,\,j\leq X}|c_{j}|,$

where

$\mu_{X}=\max_{j\leq X}\left\\{\frac{r_{j}}{r_{M}}\right\\}.$

By property (3.1), we know that $\mu_{X}<1$ and so, for $N_{0}$ sufficiently
large and $p\in\left(q_{n}\right)_{n\geq N_{0}}$,

$|\tilde{E}(X,z)|<\frac{1}{6},$

which proves (3.35). To get (3.36), we just need to use (3.15) with
$u=\frac{1}{\eta p\,\log(p)}$,

$\frac{\left|\mathcal{B}_{\alpha,p}\left(\frac{1}{\eta
p\log(p)}\right)\right|}{2\,|c_{M}|r_{M}^{2p}\left|\mathcal{G}_{p}\left(\theta_{M}\right)\right|}\leq\frac{C_{\alpha}}{2\epsilon_{0}|c_{M}|\eta
p\,\log(p)}<\frac{1}{6},$

whenever $p\geq q_{N_{0}}$ and $N_{0}$ is sufficiently large. This completes
the proof of the claim.

∎

###  3.5 Conclusion of the argument

Finally, we return to the integral representation (3.20),

$\displaystyle\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle=-\frac{8\pi
c_{M}r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\mathcal{G}_{p}\left(\theta_{M}\right)$
$\displaystyle\times\left\\{1+\frac{\mathcal{B}_{\alpha,p}(u)}{2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)}+\tilde{E}\left(X,z\right)+\tilde{H}\left(X,z\right)\right\\}$
$\displaystyle+\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\text{Re}\left[\mathcal{A}_{\alpha,p}(u,z)\right],$
(3.37)

and we take there $p\in\left(q_{n}\right)_{N_{0}\leq n\leq N}$, where $N\geq
2N_{0}+2$ and $N_{0}$ is a sufficiently large number for which (3.35) and
(3.36) hold. Replacing $u$ by $\frac{1}{\eta q_{N}\log(q_{N})}$ in (3.37), it
follows from our choices of $N_{0}$ and $X$ in the previous claim that

$1+\frac{\mathcal{B}_{\alpha,p}(u)}{2c_{M}r_{M}^{2p}\,\mathcal{G}_{p}\left(\theta_{M}\right)}+\tilde{E}\left(X,z\right)+\tilde{H}\left(X,z\right)>\frac{1}{2},$
(3.38)

by (3.35) and (3.36). Moreover, according to Claim 3.2, namely (3.30), we also
have the inequality

$\frac{2\pi\,e^{z^{2}/8}}{2^{2p}}\,\left|\text{Re}\left[\mathcal{A}_{\alpha,p}\left(\frac{1}{\eta\,p\log(p)},z\right)\right]\right|<\frac{1}{4}\cdot\frac{8\pi|c_{M}|\,r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\left|\mathcal{G}_{p}\left(\theta_{M}\right)\right|,$
(3.39)

for any $\eta\geq\eta_{0}$ and $\eta_{0}$ sufficiently large (such that (3.34)
holds). Combining (3.38) with (3.39) one concludes that, for any
$p\in\left(q_{n}\right)_{N_{0}\leq n\leq N}$ and $\eta_{0}$ chosen as in the
proof of Claim 3.2, the sequence of moments

$\left(M_{p}\right)_{p\in(q_{n})_{N_{0}\leq n\leq
N}}:=\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-\frac{2}{\eta_{0}q_{N}\log(q_{N})}\right)t\right)\,dt$
(3.40)

must have the same sign as the sequence defined by

$\left(s_{p}\right)_{p\in(q_{n})_{N_{0}\leq n\leq N}}:=-\frac{8\pi
c_{M}r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\mathcal{G}_{p}(\theta_{M}).$
(3.41)

Moreover, from (3.38) and (3.39), the modulus of $M_{p}$, (3.40), satisfies
the inequality

$\displaystyle\left|\intop_{0}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-\frac{2}{\eta_{0}q_{N}\log(q_{N})}\right)t\right)\,dt\right|$
$\displaystyle>\frac{1}{4}\cdot\frac{8\pi|c_{M}|r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\left|\mathcal{G}_{p}(\theta_{M})\right|$
$\displaystyle=\frac{2\pi|c_{M}|r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\left|\mathcal{G}_{p}(\theta_{M})\right|.$
(3.42)

In order to be able to finally apply Fejér’s lemma, we establish the following
simple bound.

######  Claim 3.4.

There exists $\eta_{1}>\eta_{0}$ such that, for any $\eta\geq\eta_{1}$ and $z$
satisfying (1.22), the following inequality takes place

$\left|\,\intop_{\eta^{2}\,q_{N}^{2}\log^{2}(q_{N})}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-\frac{2}{\eta_{0}q_{N}\log(q_{N})}\right)t\right)\,dt\,\right|<\frac{2\pi|c_{M}|r_{M}^{2p}}{2^{2p}}\left(1+e^{z^{2}/8}\sinh\left(\frac{z^{2}}{8}\right)\right)\left|\mathcal{G}_{p}(\theta_{M})\right|,$
(3.43)

whenever $p\in\left(q_{n}\right)_{N_{0}\leq n\leq N}$.

###### Proof.

Using (2.12), it is simple to check that, for a sufficiently large $\eta_{1}$
and $\eta\geq\eta_{1}$, that the left-hand side of (3.43) can be bounded as
follows

$\displaystyle\intop_{\eta^{2}q_{N}^{2}\log^{2}(q_{N})}^{\infty}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-\frac{2}{\eta_{0}q_{N}\log(q_{N})}\right)t\right)\,dt$
$\displaystyle\leq
e^{-\frac{\eta^{2}}{\eta_{0}}q_{N}\log(q_{N})}\intop_{\eta^{2}q_{N}^{2}\log^{2}(q_{N})}^{\infty}t^{2q_{N}}\,\left|\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\right|\,\exp\left(\left(\frac{\pi}{2}-\frac{1}{\eta_{0}q_{N}\log(q_{N})}\right)t\right)\,dt$
$\displaystyle\ll_{\alpha,z,\Lambda}e^{-\frac{\eta^{2}}{\eta_{0}}q_{N}\log(q_{N})}\intop_{\eta^{2}q_{N}^{2}\log^{2}(q_{N})}^{\infty}t^{2q_{N}+B(\alpha)}\,\exp\left(-\frac{t}{\eta_{0}q_{N}\log(q_{N})}+|z|\sqrt{t}\right)\,\,dt$
$\displaystyle\ll_{\alpha,z,\Lambda}e^{-\frac{\eta^{2}}{\eta_{0}}q_{N}\log(q_{N})}\intop_{\eta^{2}q_{N}^{2}\log^{2}(q_{N})}^{\infty}t^{2q_{N}+B(\alpha)}\,\exp\left(-\frac{t}{2\eta_{0}q_{N}\log(q_{N})}\right)\,dt$
$\displaystyle\ll_{\alpha,z,\Lambda}\left(2\eta_{0}q_{N}\log(q_{N})\right)^{2q_{N}+B(\alpha)+1}e^{-\frac{\eta^{2}}{\eta_{0}}q_{N}\log(q_{N})}\,\left(2q_{N}+[B(\alpha)]+1\right)!\,.$

In the above inequalities, $\ll_{\alpha,z,\Lambda}$ stands for a constant
depending on $\alpha,z$ and the pair of sequences
$\left(c_{j},\lambda_{j}\right)_{j\in\mathbb{Z}\setminus\\{0\\}}$. By
Stirling’s formula, we have that the latter expression is bounded by

$\exp\left\\{-\frac{\eta^{2}}{\eta_{0}}q_{N}\log(q_{N})+\left(2q_{N}+B(\alpha)+1\right)\log\left(2\eta_{0}q_{N}\log(q_{N})\right)+\left(2q_{N}+[B(\alpha)]+1\right)\log\left(2q_{N}+[B(\alpha)]+1\right)+O\left(q_{N}\right)\right\\}$

and so it is clear that, for $\eta\geq\eta_{1}>\eta_{0}$ sufficiently large,
the dominant term in the exponential will be
$-\frac{\eta^{2}}{\eta_{0}}q_{N}\log(q_{N})$ and this proves (3.43). ∎

By the previous claim and (3.42), we found that for each
$p\in\left(q_{n}\right)_{N_{0}\leq n\leq N}$, there exists some sufficiently
large number $\eta_{1}>\eta_{0}$ such that the sequence of moments

$\left(\tilde{M}_{p}\right)_{p\in(q_{n})_{N_{0}\leq n\leq
N}}:=\intop_{0}^{\eta_{1}^{2}q_{N}^{2}\log^{2}(q_{N})}t^{2p}\,\tilde{F}_{z,\alpha}\left(\frac{\alpha}{4}+it\right)\,\cosh\left(\left(\frac{\pi}{2}-\frac{2}{\eta_{0}q_{N}\log(q_{N})}\right)t\right)\,dt$

must have the same sign as the sequence
$\left(s_{p}\right)_{p\in\left(q_{n}\right)_{N_{0}\leq n\leq N}}$ defined by
(3.41). According to Claim 3.1, $s_{\ell}$ and $s_{\ell+1}$ have always
distinct signs, and so the sequence
$\left(s_{p}\right)_{p\in\left(q_{n}\right)_{N_{0}\leq n\leq N}}$ has exactly
$N-N_{0}-1$ sign changes. By Fejér’s theorem, we then have that

$N_{\alpha,z}\left(\eta_{1}^{2}\,q_{N}^{2}\log^{2}(q_{N})\right)\geq
N-N_{0}-1>\frac{N}{2},$ (3.44)

since $N\geq 2N_{0}+2$ by hypothesis. To conclude, we just need to connect
$q_{N}$ with $N$. We divide the final argument in two cases (recall that we
are assuming that $0<\theta_{M}<\frac{\pi}{2}$).

  1. 1.

If $\theta_{M}/\pi\in\mathbb{Q}$, say $\theta_{M}=\pi\frac{a}{b}$, then the
explicit construction (3.29) shows that, for $N\geq 3$,
$N>\frac{q_{N}-q_{2}}{b}\geq\frac{q_{N}}{2b}$. Thus, (3.44) implies

$N_{\alpha,z}\left(\eta_{1}^{2}\,q_{N}^{2}\log^{2}(q_{N})\right)>\frac{q_{N}}{4b},$

proving (1.24).

  2. 2.

If $\theta_{M}/\pi\notin\mathbb{Q}$ then, by Kronecker’s lemma, the sequence
$\left(\left\\{n\,\frac{2\theta_{M}}{\pi}\right\\}\right)_{n\in\mathbb{N}}$ is
uniformly distributed on the interval $[0,1]$, which means that

$\lim_{X\rightarrow\infty}\frac{\\#\left\\{1\leq m\leq
X\,:\,2m\theta_{M}\in\left(\mathcal{I}^{+}\cup\mathcal{I}^{-}\right)\right\\}}{X}=\frac{|\mathcal{I^{+}\cup
I^{-}}|}{2\pi}=\frac{2\theta_{M}}{\pi}.$ (3.45)

Thus, taking $X=q_{N}$ in (3.45) we obtain, for $N\geq 2N_{0}+2$ sufficiently
large,

$N_{\alpha,z}\left(\eta_{1}^{2}\,q_{N}^{2}\log^{2}(q_{N})\right)>\frac{\theta_{M}}{\pi}q_{N},$

which proves (1.24). $\blacksquare$

##  4 Lemmas for the proof of Theorem 1.2

Analogously to (2.12), we can use the estimate (2.10) to prove that

$\left|\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\right|\leq\sum_{j\neq
0}\left|c_{j}\,\eta_{f}\left(\frac{k}{2}+i\,(t+\lambda_{j})\right)\text{Re}\left(\,{}_{1}F_{1}\left(\frac{k}{2}-i\,(t+\lambda_{j});\,k;\,\frac{z^{2}}{4}\right)\right)\right|\ll_{k,z}C_{\lambda}\,\sum_{j=1}^{\infty}|c_{j}|\,|t|^{B(k)}e^{-\frac{\pi}{2}|t|+|z|\sqrt{|t|}},$
(4.1)

for some exponent $B(k)$ depending on the weight $k$ (which comes from convex
estimates for $L(s,f)$). The next lemma is an analogue of Lemma 2.1. For a
proof, we refer to [[24], p. 34, eq. (2.78)].

######  Lemma 4.1.

Let $f(\tau)$ be a holomorphic cusp form of weight $k$ for the full modular
group and $a_{f}(n)$ be its Fourier coefficients. Then for $\text{Re}(x)>0$
and any $y\in\mathbb{C}$, the following transformation formula takes place

$\sum_{n=1}^{\infty}a_{f}(n)\,n^{\frac{1-k}{2}}\,e^{-2\pi
n\,x}\,J_{k-1}\left(\sqrt{2\pi
n}\,y\right)=\frac{(-1)^{k/2}\,e^{-\frac{y^{2}}{4x}}}{x}\,\sum_{n=1}^{\infty}a_{f}(n)\,n^{\frac{1-k}{2}}\,e^{-\frac{2\pi
n}{x}}\,I_{k-1}\left(\sqrt{2\pi n}\,\,\frac{y}{x}\right)$ (4.2)

or, equivalently,

$\psi_{f}(x,z)=(-1)^{k/2}\,e^{-\frac{z^{2}}{4}}x^{-k}\,\psi_{f}\left(\frac{1}{x},\,iz\right),\,\,\,\,\text{Re}(x)>0,\,z\in\mathbb{C},$
(4.3)

where $\psi_{f}(x,z)$ is the analogue of Jacobi’s $\psi-$function (1.15),

$\psi_{f}(x,z):=(k-1)!\,\left(\sqrt{\frac{\pi
x}{2}}\,z\right)^{1-k}\sum_{n=1}^{\infty}a_{f}(n)n^{\frac{1-k}{2}}\,e^{-2\pi
n\,x}\,J_{k-1}\left(\sqrt{2\pi
n\,x}\,z\right),\,\,\text{Re}(x)>0,\,\,z\in\mathbb{C}.$ (4.4)

######  Lemma 4.2.

Let $f(\tau)$ be a holomorphic cusp form of weight $k$ for the full modular
group and $\psi_{f}(x,z)$ be the generalized Jacobi’s $\psi-$function attached
to it, (4.4). Assume that $z\in\mathbb{R}$ satisfies the condition

$-\frac{\sqrt{\pi}}{3}\leq z\leq\frac{\sqrt{\pi}}{3}.$ (4.5)

Then there is an absolute constant $A$ and a constant $C$ depending only on
$k$ such that, for any $0<u<\frac{\pi}{4}$ and every $p\in\mathbb{N}_{0}$,

$\left|\frac{d^{p}}{du^{p}}\psi_{f}\left(ie^{-2iu},z\right)\right|<C\,\frac{2^{7p}p!}{u^{k+p}}e^{-\frac{A}{u}}.$
(4.6)

###### Proof.

The proof is exactly the same as in Lemma 2.2, so we only give a brief sketch.
Starting with Cauchy’s formula,

$\left[\frac{d^{p}}{du^{p}}\psi_{f}\left(ie^{-2iu},z\right)\right]_{u=u_{0}}=\frac{p!}{2\pi
i}\,\intop_{C_{\lambda
u_{0}}(u_{0})}\,\frac{\psi_{f}\left(ie^{-2iw},z\right)}{(w-u_{0})^{p+1}}\,dw,$
(4.7)

we need to find a suitable bound for $\psi_{f}\left(ie^{-2iw},z\right),\,w\in
C_{\lambda u_{0}}(u_{0})$. This is done through Lemma 4.1 above: replacing
there $x$ by $2e^{-iw}\sin(w)$ and $y$ by
$e^{i\left(\frac{\pi}{4}-w\right)}\,z$, we obtain the transformation formula

$\displaystyle\psi_{f}\left(ie^{-2iw},z\right)$
$\displaystyle=(k-1)!\,\left(\sqrt{\frac{\pi
ie^{-2iw}}{2}}\,z\right)^{1-k}\sum_{n=1}^{\infty}a_{f}(n)n^{\frac{1-k}{2}}\,e^{-2\pi
n\cdot 2e^{-iw}\sin(w)}\,J_{k-1}\left(\sqrt{2\pi nie^{-2iw}}\,z\right)$
$\displaystyle=(k-1)!\,\left(\sqrt{\frac{\pi}{2}}e^{i\left(\frac{\pi}{4}-w\right)}z\right)^{1-k}\frac{(-1)^{k/2}\,e^{-\frac{ie^{-iw}z^{2}}{8\sin(w)}}}{2e^{-iw}\sin(w)}\,\sum_{n=1}^{\infty}(-1)^{n}a_{f}(n)\,n^{\frac{1-k}{2}}\,e^{-\frac{\pi
n}{\tan(w)}}\,I_{k-1}\left(\sqrt{\frac{\pi
n}{2}}\frac{e^{i\frac{\pi}{4}}z}{\sin(w)}\right).$

From this point on, we can easily adapt the steps given in the proof of the
bounds (2.32) and (2.34). Extracting the first term of the series in the above
expression,555without any loss of generality, we suppose that $a_{f}(1)\neq
0$. If $c$ is the smallest integer such that $a_{f}(c)\neq 0$, then it would
be possible to enlarge the interval (1.25) to
$\left[-\frac{\sqrt{\pi}c}{3},\frac{\sqrt{\pi}c}{3}\right]$. we obtain

$\displaystyle\left|\psi_{f}\left(ie^{-2iw},z\right)\right|$
$\displaystyle\leq
d_{k}\,\frac{\pi^{k}e^{-\frac{z^{2}}{8}+\frac{u_{0}}{2}}}{(2u_{0})^{k}}\exp\left\\{-\frac{\pi\sin(2\text{Re}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}P_{w}^{\star}(1)\right\\}$
$\displaystyle\times\sum_{n=1}^{\infty}\left|\frac{a_{f}(n)}{a_{f}(1)}\right|\exp\left[-\frac{\pi\sin(2\text{Re}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}\left(P_{w}^{\star}(\sqrt{n})-P_{w}^{\star}(1)\right)\right],$
(4.8)

where $P_{w}^{\star}(X)$ is the real-valued polynomial

$P_{w}^{\star}\left(X\right):=X^{2}-\frac{|z|}{\sqrt{\pi}\sin(2\text{Re}(w))}\,\sqrt{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}\,X+\frac{z^{2}}{8\pi}\,\frac{\sinh(2\text{Im}(w))}{\sin(2\text{Re}(w))}.$

Note that our expression (4.8) is completely analogous to (2.34). Repeating
the steps in (2.40), we find that the series on the right-hand side of (4.8)
is uniformly bounded for any $w\in C_{0.01u_{0}}(u_{0})$ and $z$ satisfying
(4.5). The resulting constant, $M_{k}$, will only dependent on $k$.
Analogously to (2.41) and (2.42),

$\displaystyle\left|\psi_{f}\left(ie^{-2iw},z\right)\right|$
$\displaystyle\leq\frac{d_{k}M_{k}e^{\frac{\pi}{8}}\,\pi^{k}e^{-\frac{z^{2}}{8}}}{(2u_{0})^{k}}\exp\left\\{-\frac{\pi\sin(2\text{Re}(w))}{\cosh(2\text{Im}(w))-\cos(2\text{Re}(w))}P_{w}^{\star}(1)\right\\}$
$\displaystyle<\frac{d_{k}M_{k}e^{\frac{\pi}{8}}\,\pi^{k}e^{-\frac{z^{2}}{8}}}{(2u_{0})^{k}}\exp\left[-\frac{1}{u_{0}}\left\\{\frac{1-\lambda}{e^{\frac{\pi^{2}\lambda^{2}}{24}}(1+\lambda)^{2}}-\frac{\pi|z|}{2(1-\lambda)}-\frac{\pi
z^{2}\lambda
e^{\frac{\pi^{2}\lambda^{2}}{24}}}{16(1-\lambda)^{2}}\right\\}\right]$
$\displaystyle<\frac{d_{k}M_{k}e^{\frac{\pi}{8}}\,\pi^{k}e^{-\frac{z^{2}}{8}}}{(2u_{0})^{k}}\exp\left[-\frac{1}{u_{0}}\left\\{\frac{1-\lambda}{e^{\frac{\pi^{2}\lambda^{2}}{24}}(1+\lambda)^{2}}-\frac{\pi^{\frac{3}{2}}}{6(1-\lambda)}-\frac{\pi^{2}\lambda
e^{\frac{\pi^{2}\lambda^{2}}{24}}}{144(1-\lambda)^{2}}\right\\}\right].$

Thus, if we pick $\lambda=0.01$, the term on the braces is greater than
$0.03$: this implies that

$\left|\psi_{f}\left(ie^{-2iw},z\right)\right|<\frac{d_{k}M_{k}e^{\frac{\pi}{8}}\,\pi^{k}e^{-\frac{z^{2}}{8}}}{(2u_{0})^{k}}\exp\left[-\frac{0.03}{u_{0}}\right]=\frac{C}{u_{0}^{k}}\,e^{-\frac{A}{u_{0}}},$

where $C$ depends on $k$ and $A$ is an absolute constant, say $A=0.03$. Using
Cauchy’s formula (4.7) and mimicking the steps in (2.44), we find that (4.6)
must hold for every $p\in\mathbb{N}_{0}$. ∎

We finish this section with two lemmas analogous to Lemmas 2.3 and 2.4. Since
their proofs are exactly the same, we shall omit them.

######  Lemma 4.3.

Let $g:\,\mathbb{C}\longmapsto\mathbb{C}$ be an analytic function. If
$z\in\left[-\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}},\frac{1}{6}\sqrt{\frac{\pi\alpha}{2}}\right]$
then, for arbitrary $p\in\mathbb{N}_{0}$ and $0<u<\frac{\pi}{4}$, the
following relation holds

$\left|\frac{d^{2p}}{du^{2p}}\left\\{g(u)\,\psi_{f}\left(ie^{-2iu},z\right)\right\\}\right|<D\frac{2^{14p}(2p)!e^{-\frac{A}{u}}}{u^{k+2p}}\left\|g\right\|_{L^{\infty}(C_{1}(0))},$
(4.9)

where $A$ is some absolute constant and $D$ only depends on $k$.

######  Lemma 4.4.

Let $-\frac{\pi}{4}<\omega<\frac{\pi}{4}$, $p\in\mathbb{N}$ and
$\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)$ be the function defined by
(1.23). Then the following integral representations hold666Note that the
integral is written in terms of
$i^{-k/2}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)$ because it defines a real
function of $t\in\mathbb{R}$ which is even if $k\equiv 0\mod 4$ or odd if
$k\equiv 2\mod 4$.

$\intop_{0}^{\infty}i^{-\frac{k}{2}}\,t^{2p}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\,\cosh\left(2\omega
t\right)dt=\frac{2\pi\,e^{\frac{z^{2}}{4}}}{2^{2p}}\,\text{Re}\,\left(i^{-\frac{k}{2}}\,\frac{d^{2p}}{d\omega^{2p}}\left\\{\sum_{j\neq
0}c_{j}\,e^{i\omega
k-2\omega\lambda_{j}}\,\psi_{f}\left(e^{2i\omega},z\right)\right\\}\right),\,\,\,\,k\equiv
0\mod 4$ (4.10)

and

$\intop_{0}^{\infty}i^{-\frac{k}{2}}\,t^{2p+1}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\,\cosh\left(2\omega
t\right)dt=\frac{2\pi}{2^{2p}}\,e^{\frac{z^{2}}{4}}\,\text{Re}\,\left(i^{-\frac{k}{2}}\,\frac{d^{2p+1}}{d\omega^{2p+1}}\left\\{\sum_{j\neq
0}c_{j}\,e^{i\omega
k-2\omega\lambda_{j}}\,\psi_{f}\left(e^{2i\omega},z\right)\right\\}\right),\,\,\,\,k\equiv
2\mod 4,$ (4.11)

where $\psi_{f}\left(x,z\right)$ is the “cusp form analogue” of Jacobi’s
$\psi-$function (4.4).

##  5 Proof of Theorem 1.2

In this section we follow closely the author’s variant of de la Vallée
Poussin’s method [23, 5]. The details of the proof are given for the case
where $k\equiv 0\mod 4$ and a similar argument for $k\equiv 2\mod 4$ will be
devised at the end of the proof. Let $\left(\rho_{n}\right)_{n\in\mathbb{N}}$
be the sequence of zeros of odd order of $\tilde{G}_{z,f}(s)$ such that
$\text{Re}(\rho_{n})=\frac{k}{2}$. Then we can write
$\rho_{n}:=\frac{k}{2}+i\tau_{n}$, with $\tau_{n}>0$ being an increasing
sequence777Note that if $\tilde{G}_{z,f}\left(\frac{k}{2}\right)=0$, we are
excluding this real zero from the sequence.. If we show that there is some
$h>0$ such that, for infinitely many values of $n$, $\tau_{n}<h\,n^{2}$, our
Theorem 1.2 is proved. This is the case because if we choose the sequence
$T_{n}:=hn^{2}$, then we find that $N_{f,z}\left(T_{n}\right)\geq
N_{f,z}(\tau_{n})=n=\sqrt{\frac{T_{n}}{h}}$. This establishes that

$\limsup_{T\rightarrow\infty}\,\frac{N_{f,z}(T)}{\sqrt{T}}>\frac{1}{\sqrt{h}},\,\,\,\,\text{or
equivalently }N_{f,z}(T)=\Omega\left(T^{\frac{1}{2}}\right).$ (5.1)

Hence, for the sake of contradiction, let us assume that there is some $N_{0}$
such that, for every $n\geq N_{0}$ and any $h>0$, $\tau_{n}\geq h\,n^{2}$. We
will now show that there exists some $h$ large enough that this assumption is
contradicted. Indeed, if we construct the real and entire function888The
infinite product can be written because, due to Theorem 1.4 of [24],
$\tilde{G}_{z,f}(s)$ has infinitely many zeros at the critical line
$\text{Re}(s)=\frac{k}{2}$.

$\varphi_{f,z}\left(y\right)=\prod_{\ell=1}^{\infty}\left(1-\frac{y^{2}}{\tau_{\ell}^{2}}\right)=\sum_{\ell=0}^{\infty}(-1)^{\ell}a_{2\ell}\,y^{2\ell},$
(5.2)

we see that $a_{0}=1$ and, for $\ell\geq 1$,

$a_{2\ell}=\sum_{r_{1}\geq
1}\sum_{r_{2}>r_{1}}...\sum_{r_{\ell}>r_{\ell-1}}\frac{1}{\tau_{r_{1}}^{2}\cdot...\cdot\tau_{r_{\ell}}^{2}}=\sum_{1\leq
r_{1}<r_{2}<...<r_{\ell}}\frac{1}{\tau_{r_{1}}^{2}\cdot...\cdot\tau_{r_{\ell}}^{2}},$
(5.3)

where we are summing over $(r_{1},...,r_{\ell})\in\mathbb{N}^{\ell}\text{ such
that }r_{1}<r_{2}<...<r_{\ell}$. Note that, in the $k^{\text{th}}$ nested
series in (5.3), the index $r_{k}$ always satisfies $r_{k}\geq k$, due to the
condition $r_{k}>r_{k-1}>...>r_{1}\geq 1$. From this point on, we just need to
find a suitable bound for $a_{2j}$. Considering the nested sum above, we have
two possibilities: if $1\leq k\leq\ell$ and $r_{k}\geq N_{0}$, we know by the
contradiction hypothesis that $\tau_{r_{k}}^{-2}\leq\frac{r_{k}^{-4}}{h^{2}}$,
while if $1\leq r_{k}\leq N_{0}-1$, we have that
$\tau_{r_{k}}^{-2}\leq\frac{r_{k}^{-4}}{h^{\star 2}}$ where
$h^{\star}:=\min_{1\leq n\leq
N_{0}-1}\left\\{\frac{\tau_{n}}{n^{2}}\right\\}$. But then if follows from the
argument given in [[23], p. 16, eq. (4.6)] that for some $\mathscr{B}$ (only
depending on $h$ and $N_{0}$),

$a_{2\ell}\leq\frac{\mathscr{B}}{h^{2\ell}}\,\sum_{1\leq
r_{1}<r_{2}<...<r_{\ell}}\frac{1}{r_{1}^{4}\cdot...\cdot r_{\ell}^{4}}.$ (5.4)

Defining the new coefficient on the right-hand side of (5.4)

$b_{2\ell}:=\sum_{1\leq r_{1}<...<r_{\ell}}\,\frac{1}{r_{1}^{4}\cdot...\cdot
r_{\ell}^{4}},$ (5.5)

and following the steps given in [[23], p. 17], this sequence of numbers comes
from Euler’s infinite product representation of the function

$\frac{\sinh(\pi\sqrt{y})\sin(\pi\sqrt{y})}{\pi^{2}y}=\prod_{\ell=1}^{\infty}\left(1+\frac{y}{\ell^{2}}\right)\prod_{j=1}^{\infty}\left(1-\frac{y}{\ell^{2}}\right)=\prod_{j=1}^{\infty}\left(1-\frac{y^{2}}{\ell^{4}}\right):=1+\sum_{\ell=1}^{\infty}(-1)^{\ell}b_{2\ell}\,y^{2\ell}.$
(5.6)

Thus, we can get precise information about $b_{2\ell}$ (and, consequently,
about $a_{2\ell}$) by interpreting them as the coefficients of the Taylor
series for the function on the left-hand side of (5.6). A standard computation
of these coefficients gives

$b_{2\ell}:=\sum_{1\leq r_{1}<...<r_{\ell}}\,\frac{1}{r_{1}^{4}\cdot...\cdot
r_{\ell}^{4}}=\frac{2^{2\ell+1}\pi^{4\ell}}{(4\ell+2)!}\implies
a_{2\ell}\leq\mathscr{B}\,\frac{2^{2\ell+1}\pi^{4\ell}}{h^{2\ell}(4\ell+2)!}.$
(5.7)

Our proof will be concluded by seeing that (5.7) contradicts (4.10) and the
bounds found in Lemma 4.2 above. Recall that, under the assumption $k\equiv
0\mod 4$, $i^{-k/2}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)$ is an even and
real function and $\varphi_{f,z}(t)$ has the same odd zeros as this function.
Hence, by this construction,
$i^{-k/2}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\,\varphi_{f,z}(t)$ must be
a real and even function with constant sign for any $t\in\mathbb{R}$.
Considering the continuous
$Q_{f}:\,\left(0,\,\frac{\pi}{4}\right)\longmapsto\mathbb{R}$ defined by the
integral

$Q_{f}(u):=\intop_{0}^{\infty}i^{-\frac{k}{2}}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\varphi_{f,z}(t)\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt,\,\,\,\,0<u<\frac{\pi}{4},$

we then have that $|Q_{f}(u)|$ will be positive decreasing. Our proof will now
show that this cannot be true if the bound (5.7) takes place.

If we use the power series (5.2) for $\varphi_{f,z}(t)$, we see that
$Q_{f}(u)$ can be written as an infinite series of the form

$\displaystyle Q_{f}(u)$
$\displaystyle=\intop_{0}^{\infty}i^{-\frac{k}{2}}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\,\varphi_{f,z}(t)\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle=\sum_{j=0}^{\infty}(-1)^{\ell}a_{2\ell}\,\intop_{0}^{\infty}i^{-\frac{k}{2}}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\,t^{2\ell}\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt.$
(5.8)

Note that the interchange of the orders of summation and integral in (5.8)
comes from Fubini’s theorem and the bounds (5.7): indeed

$\displaystyle\intop_{0}^{\infty}\sum_{\ell=0}^{\infty}|a_{2\ell}|t^{2\ell}\,\left|\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\right|\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)$
$\displaystyle\,dt\leq\mathscr{B}\,\intop_{0}^{\infty}\,\sum_{j=0}^{\infty}\frac{2^{2\ell+1}\pi^{4\ell}}{h^{2\ell}(4\ell+2)!}t^{2\ell}\,\left|\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\right|\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle\leq
2\mathcal{\mathscr{B}}\,\intop_{0}^{\infty}\sum_{j=0}^{\infty}\frac{1}{(4\ell)!}\,\left(\frac{2\pi^{2}t}{h}\right)^{2\ell}\left|\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\right|\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle\leq
2\mathscr{B}\,\intop_{0}^{\infty}\sum_{j=0}^{\infty}\frac{1}{\ell!}\,\left(\pi\sqrt{\frac{2t}{h}}\right)^{\ell}\left|\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\right|\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle<2\mathscr{B}\,\intop_{0}^{\infty}\left|\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\right|\exp\left(\pi\sqrt{\frac{2t}{h}}+\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle\ll_{k,z,\Lambda}\,\intop_{0}^{\infty}|t|^{B(k)}\,\exp\left(-2ut+\left(\pi\sqrt{\frac{2}{h}}+|z|\right)\sqrt{t}\right)\,dt<\infty,$

where in the last step we have used the estimate (4.1). Having assured that we
can perform the operation (5.8), it now follows from Lemma 4.4 above that

$\displaystyle
Q_{f}(u)=\sum_{j=0}^{\infty}(-1)^{\ell}a_{2\ell}\,\intop_{0}^{\infty}i^{-k/2}\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\,t^{2\ell}\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt$
$\displaystyle=\sum_{\ell=0}^{\infty}(-1)^{\ell}a_{2\ell}\,\frac{2\pi}{2^{2\ell}}\,e^{\frac{z^{2}}{4}}\,\text{Re}\,\left(i^{-\frac{k}{2}}\frac{d^{2\ell}}{du^{2\ell}}\left\\{\sum_{j\neq
0}c_{j}\,e^{i\left(\frac{\pi}{4}-u\right)k-\left(\frac{\pi}{2}-2u\right)\lambda_{j}}\,\psi_{f}\left(ie^{-2iu},z\right)\right\\}\right).$
(5.9)

Using Lemma 4.3 together with the estimate for $a_{2\ell}$ (5.7), we can bound
uniformly the previous series with respect to $u$. Indeed, since the function

$g(u)=e^{i\left(\frac{\pi}{4}-u\right)k}\,\sum_{j\neq
0}c_{j}\,e^{-\left(\frac{\pi}{2}-2u\right)\lambda_{j}}$

is analytic and $\left\|g(u)\right\|_{L^{\infty}(C_{0}(1))}\leq\mathcal{M}$,
with $\mathcal{M}$ only depending on the sequence $\sum_{j\neq 0}|c_{j}|$ (see
(3.7) above), it follows from (4.9) that

$\left|\frac{d^{2\ell}}{du^{2\ell}}\left\\{g(u)\,\psi_{f}\left(ie^{-2iu},z\right)\right\\}\right|<\mathcal{D}\frac{2^{14\ell}(2\ell)!e^{-\frac{A}{u}}}{u^{k+2\ell}},$
(5.10)

for some constant $\mathcal{D}$ depending on the weight of the cusp form and
on the sequences $\left(c_{j}\right)_{j\in\mathbb{N}}$ and on an upper bound
for the sequence $\left(\lambda_{j}\right)_{j\in\mathbb{N}}$. Thus, returning
to (5.9) and invoking (5.7)

$\displaystyle|Q_{f}(u)|$
$\displaystyle<2\pi\mathcal{D}e^{\frac{z^{2}}{4}}\frac{e^{-A/u}}{u^{k}}\,\sum_{\ell=0}^{\infty}|a_{2\ell}|\frac{2^{12\ell}(2\ell)!}{u^{2\ell}}\leq
2\pi\mathscr{C}e^{\frac{z^{2}}{4}}\frac{e^{-A/u}}{u^{k}}\sum_{\ell=0}^{\infty}\,\frac{2^{14\ell+1}\pi^{4\ell}(2\ell)!}{h^{2\ell}(4\ell+2)!}\cdot\frac{1}{u^{2\ell}}$
$\displaystyle=\mathscr{C}\pi^{\frac{3}{2}}e^{\frac{z^{2}}{4}}\frac{e^{-A/u}}{u^{k}}\,\sum_{\ell=0}^{\infty}\frac{2^{10\ell}\pi^{4\ell}}{\Gamma\left(2\ell+\frac{3}{2}\right)\left(2\ell+1\right)\left(hu\right)^{2\ell}}<\mathscr{C}\pi^{\frac{3}{2}}e^{\frac{z^{2}}{4}}\frac{e^{-A/u}}{u^{k}}\,\cosh\left(\frac{32\pi^{2}}{hu}\right)$
$\displaystyle<\mathscr{C}\frac{\pi^{\frac{3}{2}}e^{\pi/36}}{u^{k}}\exp\left(-\left(A-\frac{32\pi^{2}}{h}\right)\frac{1}{u}\right),$
(5.11)

where we have used the fact that $|z|<\frac{1}{3}\sqrt{\pi}$, (1.25). From
(5.11), if we choose the constant $h$ in such a way that
$h>\frac{32\pi^{2}}{A}$, then

$\lim_{u\rightarrow
0^{+}}\left|Q_{f}(u)\right|<\mathscr{C}\pi^{\frac{3}{2}}e^{\pi/36}\,\lim_{u\rightarrow
0^{+}}u^{-k}\,\exp\left(-\left(A-\frac{32\pi^{2}}{h}\right)\frac{1}{u}\right)=0$

which contradicts the fact that $|Q_{f}(u)|$ is positive decreasing.
Consequently, we have found $h>\frac{32\pi^{2}}{A}$ such that
$\tau_{n}<h\,n^{2}$ for infinitely many values of $n$. This shows (5.1) and so
we complete the proof of our Theorem. Finally, we note that we can replace the
value of $d$ in (1.27) by an explicit constant. Using the value $A=0.03$
(calculated in the proof of Lemma 4.2) and taking $h=\frac{36\pi^{2}}{0.03}$,
we find from (5.1) that $d=\frac{1}{36\pi}$ works.

In order to consider the case where $k\equiv 2\mod 4$, the argument is the
same with a small modification. Instead of considering $Q_{f}(u)$ we will
study the similar function,

$P_{f}(u):=\intop_{0}^{\infty}i^{-\frac{k}{2}}\,t\,\tilde{G}_{z,f}\left(\frac{k}{2}+it\right)\varphi_{f,z}(t)\,\cosh\left(\left(\frac{\pi}{2}-2u\right)t\right)\,dt,\,\,\,\,0<u<\frac{\pi}{4},$

where $\varphi_{f,z}(t)$ is given by (5.2). Invoking the second integral
representation (4.11) and using the bounds for $a_{2j}$ found in (5.7), one
can prove that $|P_{f}(u)|\rightarrow 0$ as $u\rightarrow 0^{+}$, which
assures the result. $\blacksquare$

##  6 Concluding Remarks

In this paper we have presented estimates for the number of critical zeros of
the arbitrary shifted combinations

$\tilde{F}_{z,\alpha}(s):=\sum_{j\neq
0}c_{j}\,\eta_{\alpha}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\frac{\alpha}{2};\,\frac{z^{2}}{4}\right)\right\\},$
(6.1) $\tilde{G}_{z,f}(s):=\sum_{j\neq
0}c_{j}\,\eta_{f}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(k-s-i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(k-\overline{s}+i\lambda_{j};\,k;\,\frac{z^{2}}{4}\right)\right\\}.$
(6.2)

The proofs of both estimates relied on the summation formulas (2.22) and
(4.2). One may wonder how to extend our results in different directions. For
example, one may ask if we can relax the class of hypergeometric functions
appearing in the combinations (6.1) and (6.2). Note that the second parameter
of the hypergeometric functions above depends on the Dirichlet series that
preceeds them: in the first case, it is equal to $\alpha/2$ and in the second
is $k$. Therefore, one may consider the study of the critical zeros of the
functions,

$\tilde{F}_{z,\alpha}(s;\nu):=\sum_{j\neq
0}c_{j}\,\eta_{\alpha}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(\frac{\alpha}{2}-s-i\lambda_{j};\,\nu+1;\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(\frac{\alpha}{2}-\overline{s}+i\lambda_{j};\,\nu+1;\,\frac{z^{2}}{4}\right)\right\\},$
(6.3) $\tilde{G}_{z,f}(s;\nu):=\sum_{j\neq
0}c_{j}\,\eta_{f}\left(s+i\lambda_{j}\right)\,\left\\{{}_{1}F_{1}\left(k-s-i\lambda_{j};\,\nu+1;\,\frac{z^{2}}{4}\right)+\,_{1}F_{1}\left(k-\overline{s}+i\lambda_{j};\,\nu+1;\,\frac{z^{2}}{4}\right)\right\\},$
(6.4)

where $\nu$ is now a real parameter independent of $\alpha$ (in the first
case) and of $k$ in the second. We remark that it is indeed possible to get
analogues of Theorems 1.1 and 1.2 to the shifted combinations (6.3) and (6.4)
provided $\nu\geq\frac{\alpha}{2}-1$ in the first case and $\nu\geq k-1$ in
the second. However, instead of using the summation formulas (2.22) and (4.2),
we will need more general transformations. Indeed, it is not difficult to
show999The proof of this summation formula is quite standard and it follows
the same steps as the ones given in [[24], pp. 18-19]. Although our formulas
(6.5) and (6.7) do not have an intrinsic interest themselves, they can be used
to establish new summation formulas for
$\sum_{n=1}^{\infty}r_{k}(n)\,n^{\frac{\nu-\mu}{2}}\,I_{\mu}\left(Y\sqrt{n}\right)\,K_{\nu}\left(X\sqrt{n}\right)$,
where $X>Y$ and $\text{Re}(\mu),\text{Re}(\nu)>0$. Such a formula will be
given in a forthcoming investigation and it constitutes a generalization of a
formula of Berndt, Dixit, Kim and Zaharescu [4]. that, when
$\text{Re}(\nu)>-1$ and $\text{Re}(x)>0$, $y\in\mathbb{C}$, the following
summation formula takes place

$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}r_{\alpha}(n)\,n^{-\nu/2}\,e^{-\pi
nx}\,J_{\nu}(\sqrt{\pi n}y)=-\frac{y^{\nu}\pi^{\nu/2}}{2^{\nu}\Gamma(\nu+1)}$
$\displaystyle+\frac{\pi^{\nu/2}y^{\nu}e^{-\frac{y^{2}}{4x}}}{2^{\nu}\Gamma(\nu+1)x^{\alpha/2}}\,\sum_{n=1}^{\infty}\tilde{r}_{\alpha}(n)e^{-\frac{\pi}{x}\left(n+\frac{\alpha}{4}\right)}\,\Phi_{3}\left(1-\frac{\alpha}{2}+\nu;\nu+1;\frac{y^{2}}{4x},\frac{\pi
y^{2}}{4x^{2}}\left(n+\frac{\alpha}{4}\right)\right),$ (6.5)

where $\Phi_{3}(b;c;w,z)$ is the usual Humbert function,

$\Phi_{3}(b;c;\,w,z)=\sum_{k,m=0}^{\infty}\frac{(b)_{k}}{(c)_{k+m}}\frac{w^{k}z^{m}}{k!\,m!},$
(6.6)

whose series converges absolutely for any $w,z\in\mathbb{C}$. By using (6.5)
instead of (2.22), we can establish a lower bound for the number of critical
zeros of the function (6.3). Analogously, Theorem 1.2 can be extended to
shifted combinations of the form (6.4) by using the transformation formula

$\sum_{n=1}^{\infty}a_{f}(n)\,n^{-\frac{\nu}{2}}\,e^{-2\pi
nx}\,J_{\nu}\left(\sqrt{2\pi
n}\,y\right)=\frac{(-1)^{k/2}\pi^{\nu/2}y^{\nu}x^{-k}}{2^{\nu/2}\Gamma(\nu+1)}e^{-\frac{y^{2}}{4x}}\,\sum_{n=1}^{\infty}a_{f}(n)e^{-\frac{2\pi
n}{x}}\,\Phi_{3}\left(1-k+\nu;\nu+1;\frac{y^{2}}{4x},\frac{\pi
y^{2}n}{2x^{2}}\right).$ (6.7)

Note that (6.5) and (6.7) reduce to (2.22) and (4.2) when
$\nu=\frac{\alpha}{2}-1$ and $\nu=k-1$ respectively. This is the case due to
the well-known reduction formula for Humbert’s function

$\Phi_{3}\left(0;c;w,z\right)=\Gamma(c)\,z^{-\frac{c-1}{2}}I_{c-1}\left(2\sqrt{z}\right),$

which can be immediately established from the power series (6.6). Analogous
theorems can be established when the combination of confluent hypergeometric
functions ${}_{1}F_{1}\left(a;c;z\right)$ is replaced by
${}_{2}F_{2}\left(a,b;c,d;z\right)$ and Gauss’ hypergeometric function
${}_{2}F_{1}(a,b;c;z)$.

Another direction that this work can take concerns even more general shifted
combinations of completed Dirichlet series. To continue, recall again that
$\eta(s):=\pi^{-s/2}\Gamma(s/2)\,\zeta(s)$. In the paper [7], the authors have
posed the following interesting problem: if one considers a combination of
shifted products of the form

$H_{z}(s):=\sum_{m,n=1}^{\infty}c_{m,n}\,\eta\left(s+i\lambda_{m}\right)\,\eta\left(s+i\lambda_{n}\right)\,\text{Re}\left({}_{1}F_{1}\left(\frac{1-s-i\lambda_{m}}{2};\,\frac{1}{2};\,\frac{z^{2}}{4}\right)\right)\,\text{Re}\left({}_{1}F_{1}\left(\frac{1-s-i\lambda_{n}}{2};\,\frac{1}{2};\,\frac{z^{2}}{4}\right)\right),$
(6.8)

under what circumstances will $H_{z}(s)$ have infinitely many zeros on the
critical line? Even when restrict this problem to $z=0$, it seems complicated
to formulate a result covering this case. Indeed, consider the case where
$c_{m,n}:=d_{m}\,\delta_{m,n}$, $d_{m}\geq 0$ and $z=0$: then we are dealing
with the study of the zeros of the function

$\mathcal{H}(s)=\sum_{m=1}^{\infty}d_{m}\,\eta^{2}\left(s+i\lambda_{m}\right),\,\,\,\,d_{m}\geq
0.$ (6.9)