# Correlation of shifted values of $L$-functions in the hyperelliptic ensemble

Pranendu Darbar  Indian Statistical Institute
Kolkata
West Bengal 700108, India<EMAIL_ADDRESS>and  Gopal Maiti
Indian Statistical Institute
Kolkata
West Bengal 700108, India<EMAIL_ADDRESS>
###### Abstract.

The moments of quadratic Dirichlet $L$-functions over function fields have
recently attracted much attention with the work of Andrade and Keating. In
this article, we establish lower bounds for the mean values of the product of
quadratic Dirichlet $L$-functions associated with hyperelliptic curves of
genus $g$ over a fixed finite field $\mathbb{F}_{q}$ in the large genus limit.
By using the idea of A. Florea [14], we also obtain their upper bounds. As a
consequence, we find upper bounds of its derivatives. These lower and upper
bounds give the correlation of quadratic Dirichlet $L$-functions associated
with hyperelliptic curves with different transitions.

###### Key words and phrases:

Finite fields, Function fields, Correlations of quadratic Dirichlet
$L$-functions, Hyperelliptic curves

††2010 Mathematics Subject Classification: 11G20, 11T06, 11M50.

##  1\. Introduction

The correlation of $L$-functions i.e., study of the mean values of product of
shifted values of $L$-functions near the critical line has become central to
number theory. Random matrix theory has recently become a fundamental tool for
understanding the correlation of $L$-functions. Montgomery [22] showed that
two-point correlations between the non-trivial zeros of the Riemann
$\zeta$-function, on the scale of the mean zero spacing, are similar to the
corresponding correlations between the eigenvalues of random unitary matrices
in the limit of large matrix size and conjectured that these correlations are,
in fact, identical to each other.

Keating and Snaith [21] suggested that the value distribution of the Riemann
zeta function on its critical line is related to that the characteristic
polynomials of random unitary matrices. Conjectures for the moments of
$L$-functions have been attempted for many decades, with very little progress
until the random matrix theory came into the subject.

The main observation is that the structure of the mean values of L-functions
is more clearly revealed if one considers the average of a product of
$L$-functions, where each $L$-function is evaluated at a location slightly
shifted from the critical point.

In this article, we discus about the moments and correlation of Riemann zeta
function (belonging to unitary family), Dirichlet $L$-functions (contained in
the symplectic family) and quadratic Dirichlet $L$-functions associated with
hyperelliptic curves of large genus over a fixed finite field which are also
members of the symplectic family. These families and their random matrix
analogs have been discussed from the perspective of the leading terms in the
asymptotic expressions by several authors (see [8], [9], [10], [3], [20],
[21], [11] and [19]).

Our main goal of this article is to establish lower and upper bounds for the
correlation of shifted values of quadratic Dirichlet $L$-functions near the
critical line associated to the hyperelliptic curves of large genus over a
fixed finite field.

###  1.1. Moments of the Riemann Zeta function

A classical question in the theory of Riemann zeta function is to determine
the asymptotic behaviour of

$M_{k}(T):=\int_{1}^{T}|\zeta(\tfrac{1}{2}+it)|^{2k}\,dt,$

where $k\in\mathbb{C}$, as $T\to\infty$. It is believed that for a given
positive real number $k$,

$\displaystyle{M_{k}(T)\sim c_{k}T(\log T)^{k^{2}}},$

where $c_{k}$ is a positive constant. Ramachandra [27] showed that

$M_{k}(T)\gg T(\log T)^{k^{2}},$

for any $k\in\mathbb{N}$. Using moments of characteristic polynomials of
random matrices, Keating and Snaith [20], conjectured an exact value of
$c_{k}$ for $\Re(k)>-\frac{1}{2}$. Assuming the Riemann hypothesis (RH),
Soundararajan [33] showed that for every positive real number $k$ and
$\varepsilon>0$,

(1.1) $\displaystyle M_{k}(T)\ll_{k,\varepsilon}c_{k}T(\log
T)^{k^{2}+\varepsilon}$

and Harper [17] removed the exponent $\epsilon$ in the bound of (1.1).

A generalization of the moments of $\zeta(s)$ are the shifted moments, defined
as

$M_{\bm{k}^{(m)}}(T,\bm{\alpha}^{(m)}):=\int_{0}^{T}|\zeta(\tfrac{1}{2}+it+i\alpha_{1})|^{2k_{1}}\ldots|\zeta(\tfrac{1}{2}+it+i\alpha_{m})|^{2k_{m}}\,dt,$

where $\bm{k}^{(m)}=(k_{1},\ldots,k_{m})$ is a sequence of positive real
numbers and
$\bm{\alpha}^{(m)}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{R}^{m}$ with
$\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. In [7], Chandee obtained lower and
upper bounds of $M_{\bm{k}^{(m)}}(T,\bm{\alpha}^{(m)})$ for some special
choices of $\bm{\alpha}^{(m)}$ and for a large values of $T$. More precisely,
she proved assuming the RH,

$M_{\bm{k}^{(m)}}(T,\bm{\alpha}^{(m)})\ll_{\bm{k}^{(m)},\varepsilon}T\,(\log
T)^{k_{1}^{2}+\ldots+k_{m}^{2}+\varepsilon}\prod_{i<j}\left(\min\Big{\\{}\tfrac{1}{|\alpha_{i}-\alpha_{j}|},\log
T\Big{\\}}\right)^{2k_{i}k_{j}}$

and unconditionally

$\displaystyle
M_{\bm{k}^{(m)}}(T,\bm{\alpha}^{(m)})\gg_{\bm{k}^{(m)},\bm{\alpha}^{(m)}}T\,(\log
T)^{k_{1}^{2}+\ldots+k_{m}^{2}}\prod_{i<j}\left(\min\Big{\\{}\frac{1}{|\alpha_{i}-\alpha_{j}|},\log
T\Big{\\}}\right)^{2k_{i}k_{j}},$

for sufficiently large $T$.

The moments of the derivatives of the Riemann zeta function were studied by
several mathematicians. An analog of Soundararajan’s estimate (1.1) for the
derivatives of Riemann zeta function was obtained by Milinovich [24]. Under
the RH, he showed that for every $\varepsilon>0$,

$\int_{1}^{T}|\zeta^{(l)}(\tfrac{1}{2}+it)|^{2k}\,dt\ll_{k,l,\varepsilon}T\,(\log
T)^{k^{2}+2kl+\varepsilon},$

where $k,l\in\mathbb{N}$ and $\zeta^{(l)}$ is the $l$-th derivative of
$\zeta$.

###  1.2. Moments of quadratic Dirichlet $L$-functions

Let $\chi_{d}$ be a real primitive Dirichlet character modulo $d$ given by the
Kronecker symbol $\chi_{d}(n)=\left(\tfrac{d}{n}\right)$. It is interesting to
determine the asymptotic behaviour of $\textstyle{\sum_{0<d\leq
D}L(\tfrac{1}{2},\chi_{d})^{k}}\,\,\,$ as $D\rightarrow\infty$. Extending
their approach to the zeta function, using random matrix theory, Keating and
Snaith [21] made the following conjecture about the asymptotic behaviour of
moments of Dirichlet $L$-functions $L(\frac{1}{2},\chi_{d})$.

######  Conjecture (Keating, Snaith).

For $k$ fixed with $\Re(k)\geq 0$, as $D\rightarrow\infty$

$\displaystyle\frac{1}{D}\sideset{}{{}^{\star}}{\sum}_{|d|\leq
D}L(\tfrac{1}{2},\chi_{d})^{k}\sim c_{k}\,\left(\log
D\right)^{\frac{k(k+1)}{2}},$

where $c_{k}$ is a positive constant and
$\normalsize\sideset{}{{}^{\star}}{\sum}$ indicates that the sum is over
fundamental discriminants.

In [30], Rudnick and Soundararajan obtained that for any rational number
$k\geq 1$,

$\displaystyle\frac{1}{D}\sideset{}{{}^{\star}}{\sum}_{|d|\leq
D}L(\tfrac{1}{2},\chi_{d})^{k}\gg_{k}\,\left(\log
D\right)^{\frac{k(k+1)}{2}}.$

Assuming the generalized Riemann hypothesis (GRH), Soundararajan established
that for any positive real number $k$ and $\epsilon>0$,

$\displaystyle\frac{1}{D}\sideset{}{{}^{\star}}{\sum}_{|d|\leq
D}L(\tfrac{1}{2},\chi_{d})^{k}\ll_{k,\epsilon}\,\left(\log
D\right)^{\frac{k(k+1)}{2}+\varepsilon}.$

In general, it is important to find asymptotic behaviour of the following
correlation of shifted values of Dirichlet $L$-functions:

$\displaystyle
S_{\bm{k}^{(m)}}(\bm{\alpha}^{(m)},D):=\sideset{}{{}^{\star}}{\sum}_{d\leq
D}L(\tfrac{1}{2}+\alpha_{1},\chi_{d})^{k_{1}}\dots
L(\tfrac{1}{2}+\alpha_{m},\chi_{d})^{k_{m}},$

where $\bm{k}^{(m)}=(k_{1},\dots,k_{m})$ be a sequence of real numbers and
$\bm{\alpha}^{(m)}=(\alpha_{1},\dots,\alpha_{m})$ be a sequence of complex
numbers with $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$.

Conrey $et.\,al$. [9] gave a conjecture on the asymptotic behaviour of
$S_{\bm{k}^{(m)}}(\bm{\alpha}^{(m)},D)$. Analogous questions for higher degree
$L$-functions have been studied by Milinovich and Turnage-Butterbaugh [26].

###  1.3. Moments of $L$-functions in the hyperelliptic essemble

Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and
$\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable
$t$. Let $D\in\mathbb{F}_{q}[t]$ be a monic square-free polynomial. The
quadratic character $\chi_{D}$ attached to $D$ is defined using quadratic
residue symbol for $\mathbb{F}_{q}[t]$ by
$\chi_{D}(f)=\left(\frac{D}{f}\right)$ and the corresponding Dirichlet
$L$-function is denoted by $L(s,\chi_{D})$. It is often convenient to work
with the equivalent $L$-function $\mathcal{L}(u,\chi_{D})$ written in terms of
the variable $u=q^{-s}$.

Define hyperelltiptic ensemble $\mathcal{H}_{n,q}$ or simply $\mathcal{H}_{n}$
as

$\mathcal{H}_{n}=\left\\{D\in\mathbb{F}_{q}[t]:\,D\text{ is monic, square
free, and}\,\deg(D)=n\right\\}.$

For each $D$ in the Hyperelliptic essemble $\mathcal{H}_{n}$, there is an
associated hyperelliptic curve given by $C_{D}\,:\,y^{2}=D(t)$. These curve
are non-singular and of genus $g$ given by

(1.2) $\displaystyle 2g=n-1-\lambda,$

where

$\displaystyle\lambda=\left\\{\begin{array}[c]{ll}1,&\,\text{if\, $n$
even},\\\ 0,&\,\text{if\, $n$ odd}.\end{array}\right.$

Note that, $g\to\infty$ as $n$ does so. See section 2 for more details about
the properties of Dirichlet $L$-function $L(s,\chi_{D})$ and their spectral
interpretation.

Andrade and Keating [3] conjectured that as $g\to\infty$,

(1.3)
$\displaystyle\sum_{D\in\mathcal{H}_{2g+1}}L(\tfrac{1}{2},\chi_{D})^{k}\,=\,q^{2g+1}\,\left(P_{k}(2g+1)\,+\,o(1)\right),$

where $P_{k}$ is a polynomial of degree $\frac{k(k+1)}{2}$. Assuming $q\equiv
1\left(\mathrm{mod}\,4\right)$, the conjecture (1.3) is known for $k=1$ from
the work of Andrade [2], and the error term in the asymptotic formula was
improved by Florea [12]. In [13, 14], Florea also proved the conjecture (1.3)
for $k=2,3$ and $4$ assuming $q\equiv 1\left(\mathrm{mod}\,4\right)$. For
$n=2g+2$, Jung [18] obtained that

$\frac{1}{|\mathcal{H}_{2g+2}|}\sum_{D\in\mathcal{H}_{2g+2}}L(\tfrac{1}{2},\chi_{D})=P(1)(g+1)+\frac{P^{\prime}(1)}{\log
q}-P(1)\zeta_{\mathbb{A}}\left(\frac{1}{2}\right)+O\left(2^{g+1}q^{-\frac{g}{2}}\right),$

where $P(s)=\prod_{P}\left(1-(1+|P|)^{-1}|P|^{-s}\right)$ and
$\zeta_{\mathbb{A}}\left(\frac{1}{2}\right)$ is defined in Section $2$.

Andrade [1] established the following lower bound:

######  Theorem 1.1 (Andrade).

For every even natural number $k$, we have

$\frac{1}{|\mathcal{H}_{n}|}\sum_{D\in\mathcal{H}_{n}}L(\tfrac{1}{2},\chi_{D})^{k}\gg_{k}\;n^{\tfrac{k(k+1)}{2}}.$

On the other hand, A. Florea [[14], Theorem $2.7$] found the following upper
bound for a single shifted $L$-function associated with hyperelliptic curves:

######  Theorem 1.2 (Florea).

Let $v=e^{i\theta}$, with $\theta\in[0,\pi)$. Then for every positive $k$ and
any $\epsilon>0$,

$\displaystyle\sum_{D\in\mathcal{H}_{2g+1}}\big{|}\mathcal{L}\Big{(}\frac{v}{\sqrt{q}},\chi_{D}\Big{)}\big{|}^{k}\ll_{k,\varepsilon}q^{2g+1}\;g^{\varepsilon}\,\exp\left(k\,\mathcal{M}(v,g)\,+\,\frac{k^{2}}{2}\mathcal{V}(v,g)\right),$

where
$\mathcal{M}(v,g)=\frac{1}{2}\log\left(\min\\{g,\frac{1}{2\theta}\\}\right)\quad$
and $\quad\mathcal{V}(v,g)=\mathcal{M}(v,g)+\frac{1}{2}\log g$.

####  1.3.1. Shifted moments and main results

In this article, for fixed $m$-tuple
$\bm{k}^{(m)}=(k_{1},\ldots,k_{m})\in\mathbb{N}^{m}$, we shall investigate the
following mean values of the product of $m$-shifted quadratic Dirichlet
$L$-functions:

(1.4)
$\displaystyle\mathcal{S}_{n}(\bm{v}^{(m)},\bm{k}^{(m)}):=\sum_{D\in\mathcal{H}_{n}}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}^{2k_{1}}\ldots\mathcal{L}\Big{(}\frac{v_{m}}{q^{\frac{1}{2}+\alpha_{m}}},\chi_{D}\Big{)}^{2k_{m}},$

where $\bm{v}^{(m)}=(v_{1},\ldots,v_{m})\in\mathbb{C}^{m}$ with
$v_{j}=e^{i\theta_{j}},\,\theta_{j}\in(0,\pi]$ and
$\alpha_{j}\in[0,\frac{1}{2})$ for $j=1,\ldots,m$. Also,
$\theta_{j}=\theta_{j}(g)$ is a real valued function of $g$ such that
$\displaystyle\lim_{g\to\infty}g|\theta_{j}|$ and for $i\neq j$,
$\displaystyle\lim_{g\to\infty}g|\theta_{i}-\theta_{j}|$ exists or equals
$\infty$. Note that one can obtain the moments of
$\mathcal{L}\Big{(}\frac{v}{\sqrt{q}},\chi_{D}\Big{)}$ by allowing the shifts
$\alpha_{j}$ to tend to $0$.

Throughout the article, we follow that $n$ and $g$ are connected via (1.2).
Before stating our main results, let us define

(1.5)
$\displaystyle\mu(\bm{v}^{(m)},g)=\displaystyle\sum_{j=1}^{m}k_{j}\log\left(\min\Big{\\{}\frac{1}{2|\theta_{j}|},g\Big{\\}}\right),$
(1.6)
$\displaystyle\sigma(\bm{v}^{(m)},g)=2\left(\sum_{j=1}^{m}k_{j}^{2}\right)\log{g}\,+\,2\sum_{j=1}^{m}k_{j}^{2}\log\left(\min\Big{\\{}\frac{1}{2|\theta_{j}|},g\Big{\\}}\right)$
$\displaystyle+\,4\displaystyle\sum_{i<j}k_{i}k_{j}\left(\log\left(\min\Big{\\{}\tfrac{1}{|\theta_{i}-\theta_{j}|},g\Big{\\}}\right)+\log\left(\min\Big{\\{}\tfrac{1}{|\theta_{i}+\theta_{j}|},g\Big{\\}}\right)\right).$

For $m=2$, we set

(1.7) $\displaystyle
W=\\{j\in\\{1,2\\}:\lim_{g\to\infty}g|\theta_{j}|<\infty\\}\,\text{ and
}\,W^{c}=\\{j\in\\{1,2\\}:\lim_{g\to\infty}g|\theta_{j}|=\infty\\}.$

We also define a constant depending on $W$ and $W^{c}$ as

(1.8) $\displaystyle
c_{\bm{v}^{(2)}}=\max\left\\{\underset{g\to\infty}{\lim}\;g|\theta_{1}|,\,\underset{g\to\infty}{\lim}\;g|\theta_{2}|,\,\underset{g\to\infty}{\lim}g|\theta_{1}-\theta_{2}|,\,\underset{g\to\infty}{\lim}g|\theta_{1}+\theta_{2}|\right\\},$

where maximum is taken only over the finite entries of the set.

######  Remark 1.

If $|W|=2$ and $|W^{c}|=0$ then

$c_{\bm{v}^{(2)}}=\max\left\\{\underset{g\to\infty}{\lim}\;g|\theta_{1}|,\,\underset{g\to\infty}{\lim}\;g|\theta_{2}|,\,\underset{g\to\infty}{\lim}g|\theta_{1}-\theta_{2}|,\,\underset{g\to\infty}{\lim}g|\theta_{1}+\theta_{2}|\right\\}.$

If $W=\\{1\\}$ and $W^{c}=\\{2\\}$ then
$c_{\bm{v}^{(2)}}=\lim_{g\to\infty}g|\theta_{1}|$. If $W=\emptyset$ and
$\displaystyle\lim_{g\to\infty}g|\theta_{1}-\theta_{2}|<\infty$, then
$c_{\bm{v}^{(2)}}=\displaystyle\lim_{g\to\infty}g|\theta_{1}-\theta_{2}|$.
When none of the limit is finite then $c_{\bm{v}^{(2)}}$ is an absolute
constant.

We obtain a lower bound (of the conjectured order of magnitude222The
conjectural order of magnitude of these $L$-functions in the hyperelliptic
ensemble can be compared with the autocorrelation of the random matrix
polynomials (for example, see [[10], Eqs. $(3.6)$ and $(4.19)$]). ) for
$\mathcal{S}_{n}(\bm{v}^{(2)},\bm{k}^{(2)})$ in the large degree limit i.e.
when $n$ is sufficiently large and $q$ is fixed.

######  Theorem 1.3.

Let $\bm{k}^{(2)}$ and $\bm{v}^{(2)}$ be as earlier. Assume that
$\alpha_{j}=O\left(\frac{1}{g}\right)$ for $j=1,2$. Then for $n$ large,

$\frac{1}{|\mathcal{H}_{n}|}\sum_{D\in\mathcal{H}_{n}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}\bigg{|}^{2k_{1}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{2}}{q^{\frac{1}{2}+\alpha_{2}}},\chi_{D}\Big{)}\bigg{|}^{2k_{2}}\gg_{\bm{k}^{(2)},c_{\bm{v}^{(2)}}}\exp\left(\mu\left(\bm{v}^{(2)},g\right)+\frac{1}{2}\sigma\left(\bm{v}^{(2)},g\right)\right),$

where $\mu\left(\bm{v}^{(2)},g\right)$, $\sigma\left(\bm{v}^{(2)},g\right)$
and $c_{\bm{v}^{(2)}}$ are defined by (1.5), (1.6) and (1.8) respectively.

In this article, we will provide the complete proof of the Theorem 1.3 and
from observations in the footnotes [3, 6, 8], one can easily extend Theorem
1.3 to the following form.

######  Theorem 1.4.

Let $\bm{k}^{(m)}$ and $\bm{v}^{(m)}$ be as earlier. Assume that
$\alpha_{j}=O\left(\frac{1}{g}\right)$ for all $j$. Then for $n$ large,

$\frac{1}{|\mathcal{H}_{n}|}\sum_{D\in\mathcal{H}_{n}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}\bigg{|}^{2k_{1}}\ldots\bigg{|}\mathcal{L}\Big{(}\frac{v_{m}}{q^{\frac{1}{2}+\alpha_{m}}},\chi_{D}\Big{)}\bigg{|}^{2k_{m}}\gg_{\bm{k}^{(m)},\bm{v}^{(m)}}\exp\left(\mu\left(\bm{v}^{(m)},g\right)+\frac{1}{2}\sigma\left(\bm{v}^{(m)},g\right)\right),$

where $\mu\left(\bm{v}^{(m)},g\right)$ and $\sigma\left(\bm{v}^{(m)},g\right)$
are defined by (1.5) and (1.6) respectively.

We also establish an upper bound of nearly the conjectured order of magnitude
for the sum $\mathcal{S}_{n}(\bm{v}^{(m)},\bm{k}^{(m)})$.

######  Theorem 1.5.

With the assumption as in the Theorem 1.4, for any $\epsilon>0$,

$\frac{1}{|\mathcal{H}_{n}|}\sum_{D\in\mathcal{H}_{n}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}\bigg{|}^{2k_{1}}\ldots\bigg{|}\mathcal{L}\Big{(}\frac{v_{m}}{q^{\frac{1}{2}+\alpha_{m}}},\chi_{D}\Big{)}\bigg{|}^{2k_{m}}\ll_{\bm{k}^{(m)},\varepsilon}n^{\varepsilon}\exp\left(\mu\left(\bm{v}^{(m)},g\right)+\frac{1}{2}\sigma\left(\bm{v}^{(m)},g\right)\right),$

where $\mu\left(\bm{v}^{(m)},g\right)$ and $\sigma\left(\bm{v}^{(m)},g\right)$
are defined by (1.5) and (1.6) respectively.

Theorem 1.5 is also true for any fixed sequence $\bf{k}^{(m)}$ of positive
real numbers. Let us define $\mathcal{L}^{(l)}(u,\chi_{D})$ as the $l$-th
derivative of $\mathcal{L}(u,\chi_{D})$. As an important consequence of
Theorem 1.5, we have the following upper bound.

######  Theorem 1.6.

Let $l\in\mathbb{N}$ and $\varepsilon>0$. For $n$ large, we have

$\sum_{D\in\mathcal{H}_{n}}\big{|}\mathcal{L}^{(l)}\Big{(}q^{-1/2},\chi_{D}\Big{)}\big{|}^{k}\ll_{k,l,\varepsilon}|\mathcal{H}_{n}|g^{\frac{1}{2}k(k+1)+lk+\varepsilon}.$

####  1.3.2. Applications

From the Theorem 1.5 and in light of (2.5) if we specialize $n$ as $n=2g+1$
and $\alpha_{j}$’s are zero then we recover Theorem 1.2.

The case of the mean value for $\mathcal{L}(q^{-1/2},\chi_{D})$ taken over
$\mathcal{H}_{2g+2}$ was investigated by Jung [18]. Taking $n=2g+2$, we have
the following corollary which generalizes the Theorem 1.2 :

######  Corollary 1.7.

Let $\varepsilon>0$. For $n$ large, we have

$\displaystyle\frac{1}{|\mathcal{H}_{2g+2}|}\sum_{D\in\mathcal{H}_{2g+2}}|\mathcal{L}(q^{-1/2},\chi_{D})|^{k}\ll_{k,\varepsilon}g^{\frac{1}{2}k(k+1)+\varepsilon}.$

Similarly, Theorem 1.6 provide upper bound for $k$-th moment of
$\mathcal{L}^{(m)}(q^{-1/2},\chi_{D})$ with $D\in\mathcal{H}_{2g+1}$ and
$D\in\mathcal{H}_{2g+2}$. More precisely,

######  Corollary 1.8.

Let $l\in\mathbb{N}$ and $\varepsilon>0$. For $n$ large, we have

$\sum_{D\in\mathcal{H}_{2g+1}}\big{|}\mathcal{L}^{(l)}\Big{(}q^{-1/2},\chi_{D}\Big{)}\big{|}^{k}\ll_{k,l,\varepsilon}q^{2g+1}g^{\frac{1}{2}k(k+1)+lk+\varepsilon},$
$\sum_{D\in\mathcal{H}_{2g+2}}\big{|}\mathcal{L}^{(l)}\Big{(}q^{-1/2},\chi_{D}\Big{)}\big{|}^{k}\ll_{k,l,\varepsilon}q^{2g+2}g^{\frac{1}{2}k(k+1)+lk+\varepsilon}.$

Theorem 1.4 and 1.5 can be related to the main results of [7] and [25]. In
fact we can say that
$\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}$
and
$\mathcal{L}\Big{(}\frac{v_{2}}{q^{\frac{1}{2}+\alpha_{2}}},\chi_{D}\Big{)}$
are essentially correlated when $|\theta_{j}|\asymp\frac{1}{g}$ for $j=1,2$
and independent when one of $\theta_{j}$’s is much larger than $\frac{1}{g}$.
More precisely, we have the following corollaries.

######  Corollary 1.9.

Let $W$ and $W^{c}$ be defined by (1.7). For every $\varepsilon>0$ and $n$
large,

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}\mathcal{L}\Big{(}\frac{v_{2}}{q^{\frac{1}{2}+\alpha_{2}}},\chi_{D}\Big{)}\bigg{|}^{2k}$
$\displaystyle\ll_{k,\varepsilon}\left\\{\begin{array}[c]{ll}|\mathcal{H}_{n}|\;\,{g}^{2k(4k+1)+\varepsilon}&\text{if
\; }|W|=2,\vspace{1mm}\\\
\frac{|\mathcal{H}_{n}|\,\,{g}^{3k^{2}+k+\varepsilon}}{|\theta_{2}|^{k^{2}+k}|\theta_{1}-\theta_{2}|^{2k^{2}}|\theta_{1}+\theta_{2}|^{2k^{2}}}&\text{if
\; }W=\\{1\\},W^{c}=\\{2\\},\vspace{2mm}\\\
\frac{|\mathcal{H}_{n}|\,\,{g}^{2k^{2}+\varepsilon}}{|\theta_{1}\theta_{2}|^{k^{2}+k}|\theta_{1}-\theta_{2}|^{2k^{2}}|\theta_{1}+\theta_{2}|^{2k^{2}}}&\text{if
\,}|W^{c}|=2,\,\displaystyle\lim_{g\to\infty}g|\theta_{1}-\theta_{2}|=\infty,\vspace{2mm}\\\
\frac{|\mathcal{H}_{n}|\,\,{g}^{4k^{2}+\varepsilon}}{|\theta_{1}\theta_{2}|^{k^{2}+k}|\theta_{1}+\theta_{2}|^{2k^{2}}}&\text{if
\,}|W^{c}|=2,\,\displaystyle\lim_{g\to\infty}g|\theta_{1}-\theta_{2}|<\infty,\end{array}\right.$

######  Corollary 1.10.

With the assumption as in Corollary 1.9, for $n$ large, we have

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}\mathcal{L}\Big{(}\frac{v_{2}}{q^{\frac{1}{2}+\alpha_{2}}},\chi_{D}\Big{)}\bigg{|}^{2k}$
$\displaystyle\gg_{k,\bm{u}^{(2)}}\left\\{\begin{array}[c]{ll}|\mathcal{H}_{n}|\;\,{g}^{2k(4k+1)+\varepsilon}&\text{if
\; }|W|=2,\vspace{1mm}\\\
\frac{|\mathcal{H}_{n}|\,\,{g}^{3k^{2}+k+\varepsilon}}{|\theta_{2}|^{k^{2}+k}|\theta_{1}-\theta_{2}|^{2k^{2}}|\theta_{1}+\theta_{2}|^{2k^{2}}}&\text{if
\; }W=\\{1\\},W^{c}=\\{2\\},,\vspace{2mm}\\\
\frac{|\mathcal{H}_{n}|\,\,{g}^{2k^{2}+\varepsilon}}{|\theta_{1}\theta_{2}|^{k^{2}+k}|\theta_{1}-\theta_{2}|^{2k^{2}}|\theta_{1}+\theta_{2}|^{2k^{2}}}&\text{if
\,}|W^{c}|=2,\,\displaystyle\lim_{g\to\infty}g|\theta_{1}-\theta_{2}|=\infty,\vspace{2mm}\\\
\frac{|\mathcal{H}_{n}|\,\,{g}^{4k^{2}+\varepsilon}}{|\theta_{1}\theta_{2}|^{k^{2}+k}|\theta_{1}+\theta_{2}|^{2k^{2}}}&\text{if
\,}|W^{c}|=2,\,\displaystyle\lim_{g\to\infty}g|\theta_{1}-\theta_{2}|<\infty.\end{array}\right.$

######  Remark 2.

Corollary 1.9 and 1.10 gives the lower and upper bound over all monic square-
free polynomials of both even and odd degree near the critical line.

##  2\. Background for $L$-functions over function fields

We begin this section with some preliminaries of $L$-functions over function
fields. We will use [28] as a general reference.

###  2.1. Basic facts on $\mathbb{F}_{q}[t]$

We start by fixing a finite field $\mathbb{F}_{q}$ of odd cardinality
$q=p^{r}$, $r\geq 1$ with a prime $p$. We denote by
$\mathbb{A}=\mathbb{F}_{q}[t]$ the polynomial ring over $\mathbb{F}_{q}$. For
a polynomial $f$ in $\mathbb{F}_{q}[t]$, it’s degree will be denoted by either
$\deg(f)$ or $d(f)$.

The set of all monic polynomials and monic irreducible polynomials of degree
$n$ are denoted by $\mathcal{M}_{n,q}$ (or simply $\mathcal{M}_{n}$ as we fix
$q$) and $\mathcal{P}_{n,q}$ (or simply $\mathcal{P}_{n}$) respectively. Let
$\mathcal{M}=\cup_{n\geq 1}\mathcal{M}_{n}$ and $\mathcal{P}=\cup_{n\geq
1}\mathcal{P}_{n}$. we also denote the set of all monic polynomials and monic
irreducible polynomials of degree less or equal to $n$ by $\mathcal{M}_{\leq
n,q}$ (or simply $\mathcal{M}_{\leq n}$) and $\mathcal{P}_{\leq n,q}$ (or
simply $\mathcal{P}_{\leq n}$) respectively. Let $\mathcal{H}_{n}$ denotes the
set of monic square-free polynomials of degree $n$. Observe that for $n\geq
1$, $|\mathcal{M}_{n}|=q^{n}$ and

$|\mathcal{H}_{n}|=\left\\{\begin{array}[c]{ll}q,&\;\text{if}\;n=1,\\\
q^{n-1}(q-1),&\;\text{if}\;n\geq 2.\end{array}\right.$

If $f$ is is a non-zero polynomial $\mathbb{F}_{q}[t]$, we define the norm of
$f$ to be $|f|=q^{d(f)}$. If $f=0$, we set $|f|=0$. The prime polynomial
theorem (see [28], Theorem $2.2$) states that

(2.1)
$\displaystyle|\mathcal{P}_{n,q}|=\frac{q^{n}}{n}\,\,+\,\,O\Big{(}\frac{q^{\frac{n}{2}}}{n}\Big{)}.$

The zeta function of $\mathbb{A}$, denoted by $\zeta_{\mathbb{A}}(s)$ and is
defined by

$\zeta_{\mathbb{A}}(s):=\displaystyle\sum_{f\in\mathcal{M}}\frac{1}{|f|^{s}}=\displaystyle\prod_{P\in\mathcal{P}}\left(1-|P|^{-s}\right)^{-1},\quad\quad\Re(s)>1.$

One can easily prove that $\zeta_{\mathbb{A}}(s)=\frac{1}{1-q^{1-s}}$, and
this provides an analytic continuation of zeta function to the complex plane
with a simple pole at $s=1$. Using the change of variable $u=q^{-s}$,

$\displaystyle\mathcal{Z}(u)=\sum_{f\in\mathcal{M}}u^{d(f)}=\frac{1}{1-qu},\quad\text{
if }|u|<\frac{1}{q}.$

###  2.2. Quadratic Dirichlet character and properties of their $L$-functions

For a monic irreducible polynomial $P$, the quadratic residue symbol
$\left(\frac{f}{P}\right)$ is defined by

$\displaystyle\left(\frac{f}{P}\right)=\left\\{\begin{array}[c]{ll}1,&\text{if\,
$f$ is a square\;}(\text{mod}\,\,P),\,\,P\nmid f\\\ -1,&\text{if\, $f$ is not
a square\;}(\text{mod}\,\,P),\,\,P\nmid f\\\ 0,&\text{if \; }P\mid
f.\end{array}\right.$

For monic square-free polynomial $D\in\mathbb{F}_{q}[t]$, the symbol
$\left(\frac{D}{.}\right)$ is defined by extending the above residue symbol
multiplicatively. We denote the quadratic Dirichlet character $\chi_{D}$ by

$\chi_{D}(f)=\left(\frac{D}{f}\right).$

The $L$-function associated to the quadratic Dirichlet character $\chi_{D}$ is
defined by

$\displaystyle
L(s,\chi_{D})=\sum_{f\in\mathcal{M}}\frac{\chi_{D}(f)}{|f|^{s}}=\prod_{P\in\mathcal{P}}\left(1-\chi_{D}(P)\,|P|^{-s}\right)^{-1},\;\;\Re(s)>1.$

Using the change of variable $u=q^{-s}$, we have

$\displaystyle\mathcal{L}(u,\chi_{D})=\sum_{f\in\mathcal{M}}\chi_{D}(f)\,u^{d(f)}=\prod_{P\in\mathcal{P}}\left(1-\chi_{D}(P)\,u^{d(P)}\right)^{-1},\;\;\,|u|<\frac{1}{q}.$

By [[28], Proposition $4.3$], we see that if $n\geq d(D)$ then

$\sum_{f\in\mathcal{M}_{n}}\chi_{D}(f)=0.$

It implies that $\mathcal{L}(u,\chi_{D})$ is a polynomial of degree at most
$d(D)-1$. From [29], $\mathcal{L}(u,\chi_{D})$ has a trivial zero at $u=1$ if
and only if $d(D)$ is even. This allows us to define the completed
$L$-function as

$L(s,\chi_{D})=\mathcal{L}(u,\chi_{D})=(1-u)^{\lambda}\mathcal{L}^{*}(u,\chi_{D})=(1-q^{-s})^{\lambda}{L}^{*}(s,\chi_{D}),$

where

(2.4) $\displaystyle\lambda=\left\\{\begin{array}[c]{ll}1,&\;\text{if\, $d(D)$
even},\\\ 0,&\;\text{if\, $d(D)$ odd},\end{array}\right.$

and $\mathcal{L}^{*}(u,\chi_{D})$ is a polynomial of degree

(2.5) $\displaystyle 2g=d(D)-1-\lambda$

satisfying the functional equation

$\mathcal{L}^{*}(u,\chi_{D})=(qu^{2})^{g}\mathcal{L}^{*}\left(\frac{1}{qu},\chi_{D}\right)$

Because $\mathcal{L}$ and $\mathcal{L}^{*}$ are polynomial in $u$, it is
convenient to define

$L^{*}(s,\chi_{D})=\mathcal{L}^{*}(u,\chi_{D})$

so that the above functional equation can be rewritten as

$L^{*}(s,\chi_{D})=q^{(1-2s)g}L^{*}(1-s,\chi_{D}).$

The Riemann hypothesis for curve over finite fields, established by Weil [35],
asserts that all the non-trivial zero of $\mathcal{L}^{*}(u,\chi_{D})$ are lie
on the circle $|u|=q^{-1/2}$, i.e,

$\mathcal{L}^{*}(u,\chi_{D})=\prod_{j=1}^{2g}\left(1-u\,\nu_{j}\,\right)\;\;\text{with}\;\,|\nu_{j}|=\sqrt{q}\;\,\text{for
all}\,\,j.$

One can define the completed $L$-function in the following way. Set

(2.6) $\displaystyle X_{D}(s)=|D|^{\frac{1}{2}-s}X(s),$

where

$\displaystyle
X(s)=\left\\{\begin{array}[c]{ll}q^{s-\frac{1}{2}},&\,\text{if\, $d(D)$
odd}\\\ \frac{1-q^{-s}}{1-q^{-(1-s)}}q^{-1+2s},&\,\text{if\, $d(D)$
even}.\end{array}\right.$

Let us consider

(2.7) $\displaystyle\Lambda(s,\chi_{D})=L(s,\chi_{D})X_{D}(s)^{-\frac{1}{2}}.$

Then $\Lambda(s,\chi_{D})$ satisfies the symmetric functional equation

(2.8) $\displaystyle\Lambda(s,\chi_{D})=\Lambda(1-s,\chi_{D}).$

###  2.3. Spectral Interpretation

Let $C$ be a non-singular projective curve over $\mathbb{F}_{q}$ of genus $g$.
For each extension field of degree $k$ of $\mathbb{F}_{q}$, denote by
$N_{k}(C)$ the number of points of $C$ in $\mathbb{F}_{q^{k}}$. Then, the zeta
function associated to $C$ defined as

$Z_{C}(u)=\exp\left(\sum_{k=1}^{\infty}N_{k}(C)\frac{u^{k}}{k}\right),\quad|u|<\frac{1}{q},$

is known to be a rational function of $u$ of the form

$Z_{C}(u)=\frac{P_{C}(u)}{(1-u)(1-qu)}.$

Additionally, we know that $P_{C}(u)$ is a polynomial of degree $2g$ with
integer coefficients, satisfying a functional equation

$P_{C}(u)=(qu^{2})^{g}P_{C}\left(\frac{1}{qu}\right).$

The Riemann Hypothesis, proved by Weil [35], says that the zeros of $P_{C}(u)$
all lie on the circle $|u|=\frac{1}{\sqrt{q}}$. Thus one may give a spectral
interpretation of $P_{C}(u)$ as the characteristic polynomial of a $2g\times
2g$ unitary matrix $\Theta_{C}$:

$P_{C}(u)=\det\left(I-u\sqrt{q}\Theta_{C}\right).$

Thus the eigenvalues $e^{i\theta_{j}}$ of $\Theta_{C}$ correspond to the
zeros, $q^{-1/2}e^{-i\theta_{j}}$, of $Z_{C}(u)$. The matrix $\Theta_{C}$ is
called the unitarized Frobenius class of $C$.

To put this in the context of our case, note that, for a family of
hyperelliptic curves $C_{D}:\,y^{2}=D(t)$ of genus $g$, the numerator of the
zeta function $Z_{C}(u)$ associated to $C_{D}$ is coincide with the
$L$-function $\mathcal{L}^{*}(u,\chi_{D})$, i.e.,
$P_{C}(u)=\mathcal{L}^{*}(u,\chi_{D})$.

##  3\. Preliminary Lemmas

We start with an analog of approximate functional equation for
$L(s,\chi_{D})$. Recall that $2g=n-1-\lambda$ where $\lambda$ is defined as in
(2.4).

######  Lemma 3.1 (Approximate functional equation).

Let $\chi_{D}$ be a quadratic Dirichlet character, where
$D\in\mathcal{H}_{n}$. Then for $1/2\leq s<1$,

$\displaystyle L(s,\chi_{D})$ $\displaystyle=\sum_{f\in\mathcal{M}_{\leq
g}}\frac{\chi_{D}(f)}{|f|^{s}}\;+X_{D}(s)\,\sum_{f\in\mathcal{M}_{\leq
g-1}}\frac{\chi_{D}(f)}{|f|^{1-s}}$ $\displaystyle-\lambda
q^{-s(g+1)}\sum_{f\in\mathcal{M}_{\leq
g}}\chi_{D}(f)-\lambda\,X_{D}(s)\,q^{-(1-s)g}\sum_{f\in\mathcal{M}_{\leq
g-1}}\chi_{D}(f),$

where $X_{D}(s)$ is defined by (2.6) respectively.

###### Proof.

The case $s=\frac{1}{2}$ is proved in [2] for $D\in\mathcal{H}_{2g+1}$ and
[18] for $D\in\mathcal{H}_{2g+2}$. Their methods can be easily generalized for
any $s\in(1/2,1)$. ∎

The following lemma gives an asymptotic formula for a square polynomial in
hyperelliptic ensemble.

######  Lemma 3.2.

For $f\in\mathcal{M}$, we have

$\frac{1}{|\mathcal{H}_{n}|}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(f^{2})\;=\prod_{\begin{subarray}{c}P\in\mathcal{P}\\\
P\mid
f\end{subarray}}\left(1+\frac{1}{|P|}\right)^{-1}\;+\;O(|\mathcal{H}_{n}|^{-1}).$

###### Proof.

See [[5], Lemma $3.7$] for $n=2g+1$. To get the result for $n=2g+2$, it is a
small adaptation of their proof. ∎

The following lemma is an analog of Polya-Vinogradov inequality over function
fields.

######  Lemma 3.3 (Polya-Vinogradov inequality).

For $l\in\mathcal{M}$ not a perfect square, let $l=l_{1}{l_{2}}^{2}$ with
$l_{1}$ square-free. Then for any $\epsilon>0$,

$\bigg{|}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(l)\bigg{|}\;\ll_{\varepsilon}\;\sqrt{|\mathcal{H}_{n}|}|l_{1}|^{\epsilon}.$

###### Proof.

One can easily generalize the above inequality which was proved in [[6], Lemma
$3.5$] for $n=2g+1$. Here we give a different proof in the above form for
completeness.
First assume that $l_{1}=P_{1}P_{2}\ldots P_{k}$, where $P_{j}$’s are distinct
prime polynomials, and $\deg(l_{1})\leq n$. Similar to the proof of Lemma
$3.5$ in [5], which in particular case $k=2$, one can show:

$\bigg{|}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(l)\bigg{|}=\bigg{|}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(l_{1})\bigg{|}\leq\frac{qg^{k-1}\left(d(P_{1})+\ldots+d(P_{k})\right)}{d(P_{1})\ldots
d(P_{k})}|l_{1}|^{\frac{1}{2}}\ll_{\epsilon}\sqrt{|\mathcal{H}_{n}|}|l_{1}|^{\epsilon}.$

Finally let $\deg(l_{1})>n$. We combine Lemma $3.1$ of [6] and Lemma $3.5$ of
[5] to obtain

$\bigg{|}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(l_{1})\bigg{|}\ll_{\epsilon}\sqrt{|\mathcal{H}_{n}|}|l_{1}|^{\epsilon}.$

∎

The following lemma gives an upper bound for the logarithm of
$\mathcal{L}(u,\chi_{D})$ inside the critical region.

######  Lemma 3.4.

Let $0\leq\alpha\leq\tfrac{1}{2}$, $v=e^{i\theta}$, $\theta\in[0,\pi)$ and $N$
be a positive integer. Then for $D\in\mathcal{H}_{n}$,

$\displaystyle\log{\Big{|}\mathcal{L}\Big{(}\frac{v}{q^{\frac{1}{2}+\alpha}},\chi_{D}\Big{)}\Big{|}}\leq\tfrac{2g}{N+1}\log\left(\frac{1+q^{-\alpha(N+1)}}{1+q^{-2(N+1)}}\right)+\Re\sum_{d(f)\leq
N}\frac{a_{\alpha}\left(d(f)\right)\chi_{D}(f)\,\Lambda(f)v^{d(f)}}{|f|^{\frac{1}{2}}}+O(1),$

where

$\displaystyle
a_{\alpha}(d(f))=\frac{1}{d(f)|f|^{\alpha}}-\frac{1}{d(f)|f|^{2}}\;+\;O\left(\frac{1}{(N+1)q^{(N+1)\alpha}}\right),\quad\text{
for }1\leq d(f)\leq N.$

###### Proof.

From the functional equation (2.8), we observe that

$\Big{|}\Lambda(\alpha+it,\chi_{D})\Big{|}=\bigg{|}\frac{\Lambda\left(\frac{5}{2}-it,\chi_{D}\right)\Lambda(1-\alpha-
it,\chi_{D})}{\Lambda\left(-\frac{3}{2}+it,\chi_{D}\right)}\bigg{|}=\frac{|\Lambda\left(\frac{5}{2}-it,\chi_{D}\right)||\Lambda(1-\alpha+it,\chi_{D})|}{|\Lambda\left(-\frac{3}{2}+it,\chi_{D}\right)|}.$

Recall that

$L(\alpha+it,\chi_{D})=\left(1-q^{-\alpha-
it}\right)^{\lambda}{L}^{*}(\alpha+it,\chi_{D}).$

Note that $\Big{|}{L}^{*}\left(\frac{5}{2}-it,\chi_{D}\right)\Big{|}\sim 1$.
Using the expression (2.7) for $\Lambda(s,\chi_{D})$, we get

$\displaystyle\big{|}{L}(\alpha+it,\chi_{D})\big{|}=q^{g(5-2\alpha)}\big{|}1-q^{-\alpha-
it}\big{|}^{\lambda}\prod_{j=1}^{2g}\left(\frac{q^{2\alpha-1}+1-2q^{\alpha-\frac{1}{2}}\cos(2\pi\theta_{j}-t\log{q})}{q^{4}+1-2q^{2}\cos(2\pi\theta_{j}-t\log{q})}\right)^{\frac{1}{2}},$

Since

$q^{2\alpha-1}+1-2q^{\alpha-\frac{1}{2}}\cos(2\pi\theta_{j}-t\log{q})=(q^{\alpha-\frac{1}{2}}-1)^{2}+4q^{\alpha-\frac{1}{2}}{\sin}^{2}\left(\pi\theta_{j}-\frac{t\log{q}}{2}\right)$

with a similar expression holding for the denominator, it follows that

$\displaystyle\log\left|L(\alpha+it,\chi_{D})\right|=g\left(\frac{5}{2}-\alpha\right)\log{q}-\frac{1}{2}\sum_{j=1}^{2g}\log\left(\frac{a^{2}+{\sin}^{2}(\pi\theta_{j}-\frac{t\log{q}}{2})}{b^{2}+{\sin}^{2}(\pi\theta_{j}-\frac{t\log{q}}{2})}\right)+O(1),$

where

$a=\frac{q^{2}-1}{2q},\;b=\frac{q^{\alpha-\frac{1}{2}}-1}{2q^{\frac{\alpha}{2}-\frac{1}{4}}}.$

The remaining part of the proof is the same as the proof of Lemma $8.1$ in
[14] proved by A. Florea. ∎

######  Lemma 3.5.

Let $\theta\in(-\pi,\pi)$, then we have
$\displaystyle\sum_{m=1}^{n}\frac{\cos(\theta
m)}{m}\leq\log\left(\min\Big{\\{}n,\frac{1}{|\theta|}\Big{\\}}\right)+O(1).$

###### Proof.

See [[14], Lemma $9.1$]. ∎

######  Lemma 3.6.

Let $k,\,y$ be integers such that $2ky\leq n$. For any complex numbers
$\\{a(P)\\}_{p\in\mathcal{P}}$, we have

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\bigg{|}\sum_{d(P)\leq
y}\frac{a(P)\chi_{D}(P)}{|P|^{\frac{1}{2}}}\bigg{|}^{2k}\ll\,|\mathcal{H}_{n}|\,\,\frac{(2k)!}{k!\,2^{k}}\left(\sum_{d(P)\leq
y}\frac{|a(P)|^{2}}{|P|}\right)^{k}.$

###### Proof.

This is an easy generalization of the Lemma $8.4$ of [14] and Lemma $6.3$ of
[34]. ∎

During the study of our main theorems it seems interesting to estimate the
following bounds for the zeta function over function fields. This is an analog
of bounding the Riemann zeta function near to $1$-line.

######  Lemma 3.7.

Let $v=e^{i\theta}$, where $\theta\in(-\pi,\pi)$. Let $C$ be a circle of
radius $\frac{\widetilde{r}}{g}$ centred at $\frac{1}{q}$, where

$\widetilde{r}=\lim_{g\to\infty}g\,|\theta|<\infty.$

For any $u$ in $C$, we have

$\mathcal{Z}(uv)\ll g\quad\text{if}\;\,\lim_{g\to\infty}g\,|\theta|<\infty.$

For any $u$ such that $|u-\frac{1}{q}|=O(1/g)$, we have

$\mathcal{Z}(uv)\ll\frac{1}{|\theta|}\quad\text{if}\;\,\lim_{g\to\infty}g\,|\theta|=\infty.$

###### Proof.

First assume that $\theta\in(-\pi,\,\pi)$ be such that
$\displaystyle\lim_{g\to\infty}g\,|\theta|=\infty$ . Then using
$|u-\frac{1}{q}|=O\left(1/g\right)$, we have the following estimates:

$\bigg{|}\frac{v}{(1-v)}(1-qu)\bigg{|}=o(1)$

and

$|(1-v)^{-1}|\ll\frac{1}{|\theta|}.$

Thus

$|\mathcal{Z}(uv)|=|(1-quv)^{-1}|=\bigg{|}(1-v)^{-1}\left(1+\frac{v}{(1-v)}(1-qu)\right)^{-1}\bigg{|}\ll\frac{1}{|\theta|}.$

Finally let $\theta$ be such that
$\displaystyle\lim_{g\to\infty}g\,|\theta|<\infty$. Then
$|u-\frac{1}{q}|\leq\frac{\widetilde{r}}{g}$. We use the change of variable
$u=q^{-s}$ to get the hypothesis of the form
$|s-1|\leq\frac{\widetilde{r}}{g}$. Since

$\mathcal{Z}(uv)=\sum_{f\in\mathcal{M}}(uv)^{\deg(f)},$

it is enough to show that

$\sum_{f\in\mathcal{M}}\frac{1}{|f|^{1+\,\widetilde{r}/g\,-\,i\theta/\log
q}}=O(g).$

Therefore using Lemma 3.5 and the prime polynomial theorem, we obtain

$\displaystyle\log\bigg{|}\sum_{f\in\mathcal{M}}\frac{1}{|f|^{1+\,\widetilde{r}/g\,-\,i\theta/\log
q}}\bigg{|}=$
$\displaystyle\Re\sum_{P\in\mathcal{P}}\frac{1}{|P|^{1+\,\widetilde{r}/g\,-\,i\theta/\log
q}}\,+\,O(1)=\Re{\sum_{n}\frac{1}{nq^{n(\widetilde{r}/g\,-\,i\theta/\log
q)}}}+\,O(1)$ $\displaystyle=$ $\displaystyle\Re{\sum_{n\leq
g}\frac{q^{\frac{in\theta}{\log q}}}{n}}-\Re{\sum_{n\leq
g}\left(\frac{1}{n}-\frac{1}{nq^{\frac{\widetilde{r}n}{g}}}\right)q^{\frac{in\theta}{\log
q}}}+\Re{\sum_{n>g}\frac{q^{\frac{in\theta}{\log
q}}}{nq^{\frac{\widetilde{r}n}{g}}}}+O(1)$ $\displaystyle=$
$\displaystyle{\sum_{n\leq
g}\frac{\cos(n\theta)}{n}}+O(1)\leq\;\log\left(\min\left\\{g,\frac{1}{|\theta|}\right\\}\right)\leq\;\log{g},$

and the lemma’s proof is concluded. ∎

##  4\. Proof of Theorem 1.3

Throughout this section, for the sake of simplicity, we write $\bm{v}^{(2)}$
and $\bm{k}^{(2)}$ simply as $\bm{v}$ and $\bm{k}$ respectively. For any
$k_{1},k_{2}\in\mathbb{N}$, we write

(4.1)
$\displaystyle{\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}=\sum_{f\in\mathcal{M}}a_{f}\frac{\chi_{D}(f)}{|f|^{1/2}},$

where

(4.2) $\displaystyle
a_{f}=\sum_{f_{1}f_{2}=f}\frac{\tau_{k_{1}(f_{1})}\tau_{k_{2}(f_{2})}}{|f_{1}|^{\alpha_{1}}|f_{2}|^{\alpha_{2}}}\,e^{i\big{(}\theta_{1}\,d(f_{1})+\theta_{2}\,d(f_{2})\big{)}}.$

We start by defining the following truncated $L$-function which is an analog
of Dirichlet polynomials over number fields:

$\displaystyle\mathcal{L}_{\leq(k_{1}+k_{2})X}\big{(}\bm{v},\chi_{D}\big{)}:=\sum_{f\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}a_{f}\frac{\chi_{D}(f)}{|f|^{1/2}},$

where $a_{f}$ is defined by (4.2) and the parameter $X$ will be chosen later.
We call $X$ as point of truncation of (4.1).

Using Cauchy-Schwarz inequality, we have

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\Big{|}{\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}\,\overline{\mathcal{L}_{\leq(k_{1}+k_{2})X}\big{(}\bm{v},\chi_{D}\big{)}}\Big{|}\hskip
142.26378pt$
$\displaystyle\leq\left(\sum_{D\in\mathcal{H}_{n}}\Big{|}{\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}\Big{|}^{2}\right)^{1/2}\left(\sum_{D\in\mathcal{H}_{n}}\Big{|}\overline{\mathcal{L}_{\leq(k_{1}+k_{2})X}\big{(}\bm{v},\chi_{D}\big{)}}\Big{|}^{2}\right)^{1/2}.$

Therefore, we obtain

(4.3)
$\displaystyle\sum_{D\in\mathcal{H}_{n}}\Big{|}{\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}\Big{|}^{2}\geq\;\frac{S_{1}^{2}}{S_{2}},$

where

$\displaystyle
S_{1}:=\sum_{D\in\mathcal{H}_{n}}\Big{|}{\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}\,\overline{\mathcal{L}_{\leq(k_{1}+k_{2})X}\big{(}\bm{v},\chi_{D}\big{)}}\Big{|}$

and

$\displaystyle
S_{2}:=\sum_{D\in\mathcal{H}_{n}}\Big{|}\overline{\mathcal{L}_{\leq(k_{1}+k_{2})X}\big{(}\bm{v},\chi_{D}\big{)}}\Big{|}^{2}.$

Now we establish an asymptotic formula for $S_{2}$ and a lower bound for
$S_{1}$.

###  4.1. Estimation of the sum $S_{2}$

Inserting the $D$-sum after expanding square in $S_{2}$, we get

$\displaystyle S_{2}=$
$\displaystyle\sum_{f\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\sum_{f^{\prime}\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{a_{f}\,\overline{a_{f^{\prime}}}}{|ff^{\prime}|^{1/2}}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(ff^{\prime}).$

######  Case 1.

Assume that $ff^{\prime}\neq\square$. Observe that
$a_{f}\ll_{\varepsilon}|f|^{\varepsilon}$ and using Lemma 3.3, we obtain that

$\displaystyle S_{2}\ll\sqrt{|\mathcal{H}_{n}|}\sum_{f\in\mathcal{M}_{\leq
2(k_{1}+k_{2})X}}\frac{1}{|f|^{\frac{1}{2}-\varepsilon}}\ll\sqrt{|\mathcal{H}_{n}|}\,q^{2\left(\frac{1}{2}+\varepsilon\right)(k_{1}+k_{2})X}.$

Let us choose $X=\frac{g}{2(k_{1}+k_{2})}$ 333For the Theorem 1.4, the point
of truncation will be $(k_{1}+\ldots+k_{m})X$ and the choice of $X$ is equal
to $\frac{g}{2(k_{1}+\ldots+k_{m})}$.. So, we have

$S_{2}\ll q^{\left(\frac{3}{2}+\varepsilon\right)g}.$

######  Case 2.

Assume that $ff^{\prime}=\square=l^{2}$, where $l\in\mathbb{F}_{q}[t]$. By
using Lemma 3.2 and $\tau_{k}(f)\ll_{\varepsilon}|f|^{\varepsilon}$,

$\displaystyle
S_{2}=|\mathcal{H}_{n}|\sum_{l\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{1}{|l|}\sum_{\begin{subarray}{c}f_{1}f_{2}f_{3}f_{4}=l^{2}\end{subarray}}\frac{\tau_{k_{1}(f_{1})}\tau_{k_{1}(f_{2})}\tau_{k_{2}(f_{3})}\tau_{k_{2}(f_{4})}}{|f_{1}f_{2}|^{\alpha_{1}}|f_{3}f_{4}|^{\alpha_{2}}}$
$\displaystyle\times\;e^{i\theta_{1}(d(f_{1})-d(f_{2}))+i\theta_{2}(d(f_{3})-d(f_{4}))}\;\prod_{P\mid
l}\left(1+\frac{1}{|P|}\right)^{-1}$
$\displaystyle+\;\;O\left(\sum_{l\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{1}{|l|}\sum_{\begin{subarray}{c}f_{1}f_{2}f_{3}f_{4}=l^{2}\end{subarray}}\frac{\tau_{k_{1}(f_{1})}\tau_{k_{1}(f_{2})}\tau_{k_{2}(f_{3})}\tau_{k_{2}(f_{4})}}{|f_{1}f_{2}|^{\alpha_{1}}|f_{3}f_{4}|^{\alpha_{2}}}\right)$
$\displaystyle=|\mathcal{H}_{n}|\sum_{l\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{b(l)}{|l|}+\;\;O\left(q^{\varepsilon(k_{1}+k_{2})X}\right),\hskip
56.9055pt$

where

$\displaystyle
b(l)=\sum_{\begin{subarray}{c}f_{1}f_{2}f_{3}f_{4}=l^{2}\end{subarray}}\frac{\tau_{k_{1}(f_{1})}\tau_{k_{1}(f_{2})}\tau_{k_{2}(f_{3})}\tau_{k_{2}(f_{4})}}{|f_{1}f_{2}|^{\alpha_{1}}|f_{3}f_{4}|^{\alpha_{2}}}\,e^{i\theta_{1}(d(f_{1})-d(f_{2}))+i\theta_{2}(d(f_{3})-d(f_{4}))}\prod_{P\mid
l}\left(1+\frac{1}{|P|}\right)^{-1}.$

We use the Perron’s formula444Perron’s formula in function fields comes
through the Cauchy’s integral formula. More precisely
$\displaystyle{\sum_{f\in\mathcal{M}_{\leq X}}}a_{f}=\frac{1}{2\pi
i}\underset{|u|=r}{\int}\Big{(}\displaystyle{\sum_{f\in\mathcal{M}}}a_{f}\,u^{\deg(f)}\Big{)}\frac{du}{u^{X+1}(1-u)}$,
provided that the power series
$\displaystyle{\sum_{f\in\mathcal{M}}}a_{f}\,u^{\deg(f)}$ is absolutely
convergent in $|u|\leq r<1$. to get

$\displaystyle\sum_{l\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{b(l)}{|l|}=\frac{1}{2\pi
i}\underset{|u|=r}{\int}B(u)\frac{(qu)^{-(k_{1}+k_{2})X}}{(1-qu)}\frac{du}{u},$

where

$B(u)=\displaystyle\sum_{l\in\mathcal{M}}b(l)u^{d(l)}\quad\text{ and }\quad
r<\frac{1}{q}.$

For an irreducible polynomial $P$, we observe that

$\displaystyle
b(P)=\left(1+\frac{1}{|P|}\right)^{-1}\sum_{\begin{subarray}{c}f_{1}f_{2}f_{3}f_{4}=P^{2}\end{subarray}}\frac{\tau_{k_{1}(f_{1})}\tau_{k_{1}(f_{2})}\tau_{k_{2}(f_{3})}\tau_{k_{2}(f_{4})}}{|f_{1}f_{2}|^{\alpha_{1}}|f_{3}f_{4}|^{\alpha_{2}}}\,e^{i\theta_{1}(d(f_{1})-d(f_{2}))+i\theta_{2}(d(f_{3})-d(f_{4}))}$
$\displaystyle=\left(1+\frac{1}{|P|}\right)^{-1}\left(\sum_{\begin{subarray}{c}j=1\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}^{2}\frac{k_{j}(k_{j}+1)}{2}\frac{e^{2i\epsilon_{j}\theta_{j}\,d(P)}}{|P|^{2\alpha_{j}}}+\sum_{j=1}^{2}\frac{k_{j}^{2}}{|P|^{2\alpha_{j}}}+\sum_{\epsilon_{j}\in\\{\pm
1\\}}\frac{k_{1}k_{2}}{|P|^{\alpha_{1}+\alpha_{2}}}e^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})d(P)}\right),$

which allows us to write $B(u)$ as

$B(u)=\displaystyle\prod_{j=1}^{2}\mathcal{Z}^{k_{j}^{2}}(u)\displaystyle\prod_{\begin{subarray}{c}j=1\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}^{2}\mathcal{Z}^{\frac{k_{j}(k_{j}+1)}{2}}\left(ue^{2i\epsilon_{j}\theta_{j}}\right)\displaystyle\prod_{\epsilon_{j}\in\\{\pm
1\\}}\mathcal{Z}^{k_{1}k_{2}}\left(ue^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}\right)C(u).$

Here $C(u)$ is absolutely convergent for $|u|<\frac{1}{\sqrt{q}}$. Therefore,

(4.4)
$\displaystyle\begin{split}&\sum_{l\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{b(l)}{|l|}=\frac{1}{2\pi
i}\underset{|u|=r}{\int}B(u)\frac{(qu)^{-(k_{1}+k_{2})X}}{(1-qu)}\frac{du}{u}\\\
&=\frac{1}{2\pi
i}\underset{|u|=r}{\int}\prod_{j=1}^{2}\mathcal{Z}^{k_{j}^{2}}(u)\prod_{\begin{subarray}{c}j=1\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}^{2}\mathcal{Z}^{\frac{k_{j}(k_{j}+1)}{2}}\left(ue^{2i\epsilon_{j}\theta_{j}}\right)\prod_{\epsilon_{j}\in\\{\pm
1\\}}\mathcal{Z}^{k_{1}k_{2}}\left(ue^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}\right)\\\
&\hskip 256.0748pt\times
C(u)\frac{(qu)^{-(k_{1}+k_{2})X}}{(1-qu)}\frac{du}{u},\end{split}$

where $r=\frac{1}{q^{1+\varepsilon}}$.

####  4.1.1. Calculating the main term of $S_{2}$

To get main term we have to shift the contour of integration (4.4) over $u$ to
a circle of radius $|u|=R=\frac{1}{q^{1/2\,+\,\varepsilon}}$. The integrand
has a pole at $u=\frac{1}{q}$ of order $k_{1}^{2}+k_{2}^{2}+1$ and at
$u=\frac{1}{qe^{2i\epsilon_{j}\theta_{j}}}$ of order
$\frac{k_{j}(k_{j}+1)}{2}$ and at
$u=\frac{1}{q\,e^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}}$ of order
$k_{1}k_{2}$, where $\epsilon_{j}\in\\{\pm 1\\}$ and $j=1,2$.

We define

$\displaystyle
D(u)=\prod_{j=1}^{2}\mathcal{Z}^{k_{j}^{2}}(u)\prod_{\begin{subarray}{c}j=1\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}^{2}\mathcal{Z}^{\frac{k_{j}(k_{j}+1)}{2}}\left(ue^{2i\epsilon_{j}\theta_{j}}\right)\prod_{\epsilon_{j}\in\\{\pm
1\\}}\mathcal{Z}^{k_{1}k_{2}}\left(ue^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}\right)C(u)\frac{(qu)^{-(k_{1}+k_{2})X}}{u(1-qu)}.$

Using the Cauchy’s residue theorem555Cauchy’s residue theorem says that if
$\gamma$ is a simple closed, positively oriented contour in the complex plane
and $f$ is analytic excepts for some points $z_{1},\ldots,z_{n}$ inside
$\gamma$, then $\underset{\gamma}{\oint}f(z)\,dz=2\pi
i\displaystyle\sum_{k=1}^{n}\underset{z=z_{k}}{Res}f(z)$. , we obtain

$\displaystyle\frac{1}{2\pi i}\underset{|u|=r}{\int}D(u)\,du$
$\displaystyle=\frac{1}{2\pi
i}\underset{|u|=R}{\int}D(u)\,du-\underset{u=1/q}{\mathrm{Res}}D(u)-\sum_{\begin{subarray}{c}j=1\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}^{2}\underset{u=1/qe^{2i\epsilon_{j}\theta_{j}}}{\mathrm{Res}}D(u)$
$\displaystyle-\sum_{\epsilon_{j}\in\\{\pm
1\\}}\underset{u=1/qe^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}}{\mathrm{Res}}D(u),\hskip
85.35826pt$

where $r=\frac{1}{q^{1+\varepsilon}}$ and
$R=\frac{1}{q^{1/2\,+\,\varepsilon}}$.

On the circle $|u|=R=\frac{1}{q^{1/2\,+\,\varepsilon}}$, we see that the
functions $\frac{1}{1-qu},$
$\frac{1}{1-que^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}}$ and
$\frac{1}{1-que^{2i\epsilon_{j}\theta_{j}}}$ are bounded. This leads

$\displaystyle\frac{1}{2\pi i}\underset{|u|=R}{\int}D(u)\,du\ll
q^{-(\frac{1}{2}-\varepsilon)(k_{1}+k_{2})X}.$

#### Evaluation of the sum of residues

We claim that

$\displaystyle\underset{u=1/q}{\mathrm{Res}}D(u)+\sum_{\begin{subarray}{c}j=1\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}^{2}\underset{u=1/qe^{2i\epsilon_{j}\theta_{j}}}{\mathrm{Res}}D(u)+\sum_{\epsilon_{j}\in\\{\pm
1\\}}\underset{u=1/qe^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}}{\mathrm{Res}}D(u)\hskip
113.81102pt$ $\displaystyle\hskip
5.69054pt\sim_{\bm{k},\widetilde{c}}g^{k_{1}^{2}+k_{2}^{2}}\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{2|\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\tfrac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\left(\min\Big{\\{}\tfrac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}},$

where

(4.5)
$\displaystyle\widetilde{c}:=\widetilde{c}_{\bm{v}}+1\;\text{with}\;\widetilde{c}_{\bm{v}}\;\text{is
defined as in \eqref{contant of main theorem}}.$

Let us define the following sets

$\displaystyle
W_{1}=\\{j\in\\{1,2\\}:\underset{g\to\infty}{\lim}g|\theta_{j}|<\infty\\},\quad{W}^{c}_{1}=\\{j\in\\{1,2\\}:\underset{g\to\infty}{\lim}g|\theta_{j}|=\infty\\},$
$\displaystyle
W_{2}=\\{(1,2):\underset{g\to\infty}{\lim}g|\theta_{1}-\theta_{2}|<\infty\\},\quad{W}^{c}_{2}=\\{(1,2):\underset{g\to\infty}{\lim}\,g|\theta_{1}-\theta_{2}|=\infty\\},$
$\displaystyle
W_{-2}=\\{(1,2):\underset{g\to\infty}{\lim}g|\theta_{1}+\theta_{2}|<\infty\\},\quad{W}^{c}_{-2}=\\{(1,2):\underset{g\to\infty}{\lim}g|\theta_{1}+\theta_{2}|=\infty\\}.$

We call the elements of the sets $W_{1}$ and $W^{c}_{1}$ as finite and
infinite “single shift” respectively. We also call the elements of the sets
$W_{\epsilon 2}$ and ${W}^{c}_{\epsilon 2},\,\epsilon\in\\{1,-1\\}$ as finite
and infinite “pair shift” respectively666Note that for two dimensional
correlations only one of the sets $W_{\epsilon 2},{W}^{c}_{\epsilon
2},\,\epsilon\in\\{1,-1\\}$ contains the “pair shift” $(1,2)$ but for higher
dimensional correlations either of the sets $W_{\epsilon 2},{W}^{c}_{\epsilon
2}$ may contain more than one “pair shift” which are of the form
$(j_{1},j_{2}).$.

#### Estimation of finite “single shift” and “pair shift”

Cauchy’s residue theorem allows us to write

$\displaystyle\underset{u=\frac{1}{q}}{\mathrm{Res}}D(u)+\sum_{\begin{subarray}{c}j\in
W_{1}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\underset{u=\frac{1}{qe^{2i\epsilon_{j}\theta_{j}}}}{\mathrm{Res}}D(u)+\sum_{\begin{subarray}{c}(1,2)\in
W_{\epsilon 2}\\\ \epsilon,\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\underset{u=\frac{1}{qe^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2}})}}{\mathrm{Res}}D(u)=\underset{\Gamma}{\int}D(u)\,du,$

where $\Gamma$ is a circle centered at $\frac{1}{q}$ of radius
$\frac{\widetilde{c}}{g}$ and $\widetilde{c}$ is defined by (4.5). We apply
the definition of the sets $W_{1},\,W_{1}^{c}$ and $W_{\epsilon
2},\,W_{\epsilon 2}^{c}$ to write

$\displaystyle
D(u)=\left(\frac{1}{1-qu}\right)^{k_{1}^{2}+k_{2}^{2}}\prod_{\begin{subarray}{c}j\in
W_{1}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\left(\frac{1}{1-que^{2i\epsilon_{j}\theta_{j}}}\right)^{\frac{k_{j}(k_{j}+1)}{2}}\prod_{\begin{subarray}{c}(1,2)\in
W_{\epsilon 2}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\left(\frac{1}{1-que^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}}\right)^{k_{1}k_{2}}$
$\displaystyle\times\prod_{\begin{subarray}{c}j\in{W}^{c}_{1}\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\mathcal{Z}^{\frac{k_{j}(k_{j}+1)}{2}}\left(ue^{2i\epsilon_{j}\theta_{j}}\right)\prod_{\begin{subarray}{c}(1,2)\in{W}^{c}_{\epsilon
2}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\mathcal{Z}^{k_{1}k_{2}}\left(ue^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}\right)C(u)\frac{(qu)^{-(k_{1}+k_{2})X}}{u(1-qu)}$
$\displaystyle=\left(\frac{1}{1-qu}\right)^{k_{1}^{2}+k_{2}^{2}+1}\prod_{\begin{subarray}{c}j\in
W_{1}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\left(\frac{1}{1-que^{2i\epsilon_{j}\theta_{j}}}\right)^{\frac{k_{j}(k_{j}+1)}{2}}\prod_{\begin{subarray}{c}(1,2)\in
W_{\epsilon 2}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\left(\frac{1}{1-que^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}}\right)^{k_{1}k_{2}}$
$\displaystyle\times(qu)^{-(k_{1}+k_{2})X}\,\widetilde{E}(u),\hskip
85.35826pt$

where

$\displaystyle\widetilde{E}(u)=\prod_{\begin{subarray}{c}j\in{W}^{c}_{1}\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\mathcal{Z}^{\frac{k_{j}(k_{j}+1)}{2}}\left(ue^{2i\epsilon_{j}\theta_{j}}\right)\prod_{\begin{subarray}{c}(1,2)\in{W}^{c}_{\epsilon
2}\\\ \epsilon,\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}\end{subarray}}\mathcal{Z}^{k_{1}k_{2}}\left(ue^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}\right)\frac{C(u)}{u}.$

Note that $\widetilde{E}(u)$ is analytic on and inside the circle $\Gamma$ and
it’s radius of convergence is $\gg\frac{1}{g}$. Therefore for
$|u-\frac{1}{q}|=O(\frac{1}{g})$,

$\displaystyle\widetilde{E}(u)=\sum_{n=0}^{\infty}e_{n}\left(1-qu\right)^{n}.$

Next we evaluate the integral $\displaystyle\int_{\Gamma}D(u)\,du$ which is
equal to

(4.6)
$\displaystyle\int_{\Gamma}\frac{1}{(1-qu)^{V+1}}\left(1+\sum_{n=1}^{\infty}\,\frac{b_{n}}{(1-qu)^{n}}\right)\widetilde{E}(u)\,(qu)^{-(k_{1}+k_{2})X}\,du,$

where

$\displaystyle V=k_{1}^{2}+k_{2}^{2}\,+\displaystyle\sum_{j\in
W_{1}}k_{j}(k_{j}+1)+\displaystyle\sum_{\begin{subarray}{c}(1,2)\in
W_{\epsilon 2}\\\ \epsilon\in\\{\pm 1\\}\end{subarray}}2k_{1}k_{2}\quad\text{
and }$ $\displaystyle
1+\sum_{n=1}^{\infty}\frac{b_{n}}{(1-qu)^{n}}=\prod_{\begin{subarray}{c}j\in
W_{1}\\\ \epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\left(1+\sum_{n=1}^{\infty}(-1)^{n}\binom{\frac{k_{j}(k_{j}+1)}{2}+n}{\frac{k_{j}(k_{j}+1)}{2}}\frac{(e^{-2i\epsilon_{j}\theta_{j}}-1)^{n}}{(1-qu)^{n}}\right)$
$\displaystyle\hskip 85.35826pt\times\prod_{\begin{subarray}{c}(1,2)\in
W_{\epsilon 2}\\\ \epsilon,\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}\end{subarray}}\left(1+\sum_{m=1}^{\infty}(-1)^{m}\binom{k_{1}k_{2}+m}{k_{1}k_{2}}\frac{(e^{-i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}-1)^{m}}{(1-qu)^{m}}\right).$

For $n\geq 0$, we deduce that

$\displaystyle\int_{\Gamma}\frac{1}{(1-qu)^{V+1}}\frac{b_{n}}{(1-qu)^{n}}\widetilde{E}(u)(qu)^{-(k_{1}+k_{2})X}\,du\hskip
142.26378pt$
$\displaystyle=\,e_{0}b_{n}\frac{F_{V+n}((k_{1}+k_{2})X)}{(V+n)!}\,+\sum_{l=1}^{V+n}e_{l}b_{n}\frac{F_{V+n-l}((k_{1}+k_{2})X)}{(V+n-l)!},$

where $F_{n}(x)=x(x+1)(x+2)\ldots(x+n-1)$, for $n\geq 1$ and $F_{0}(x)=1$.

From the choice of $X=\frac{g}{2(k_{1}+k_{2})}$, right hand side of the above
equation becomes

(4.7)
$\displaystyle\frac{e_{0}\,b_{n}}{(V+n)!}\Big{(}\frac{g}{2}\Big{)}^{V+n}+\sum_{l=1}^{V+n}\frac{e_{l}\,b_{n}\,d_{V+n-l}}{(V+n-l)!}\Big{(}\frac{g}{2}\Big{)}^{V+n-l},$

where

$d_{l}=1+\frac{l!}{e_{V+n-l}}\left(\frac{s^{(l)}_{l-1}e_{V+n-(l+1)}}{(l+1)!}+\frac{s^{(l+1)}_{l-1}e_{V+n-(l+2)}}{(l+2)!}+\ldots+\frac{s^{(V+n-2)}_{l-1}e_{1}}{(V+n-1)!}+\frac{s^{(V+n-1)}_{l-1}e_{0}}{(V+n)!}\right)$

with

$s^{(k)}_{k-i}=\sum_{1\leq l_{1}<\ldots<l_{i}\leq k}l_{1}\ldots
l_{i}\,,\;\;i=1,2,\ldots,k.$

The coefficients $s^{(k)}_{k-i}$ are called the Stirling numbers of first kind
and $s^{(k)}_{k-i}\leq(k+1)!$ (see [16], equation (6.9)). For more details
about $d_{l}$ see Appendix 7.2. Therefore, the integral (4.6) is equal to

(4.8)
$\begin{multlined}\frac{e_{0}g^{V}}{2^{V}}\sum_{n=0}^{\infty}\frac{b_{n}\,g^{n}}{2^{n}(V+n)!}+\sum_{l=1}^{V}\frac{e_{l}\,g^{V-l}}{2^{V-l}}\sum_{n=0}^{\infty}\frac{b_{n}\,d_{V+n-l}\,g^{n}}{2^{n}(V+n-l)!}\\\
+\sum_{l=1}^{\infty}e_{V+l}\sum_{n=0}^{\infty}\frac{b_{n+l}\,d_{n}\,g^{n}}{2^{n}n!}.\end{multlined}\frac{e_{0}g^{V}}{2^{V}}\sum_{n=0}^{\infty}\frac{b_{n}\,g^{n}}{2^{n}(V+n)!}+\sum_{l=1}^{V}\frac{e_{l}\,g^{V-l}}{2^{V-l}}\sum_{n=0}^{\infty}\frac{b_{n}\,d_{V+n-l}\,g^{n}}{2^{n}(V+n-l)!}\\\
+\sum_{l=1}^{\infty}e_{V+l}\sum_{n=0}^{\infty}\frac{b_{n+l}\,d_{n}\,g^{n}}{2^{n}n!}.$

We claim that the main contribution comes from only the first term of the
above expression. To prove this, we have to find an upper bound for the
coefficients $b_{n}$, $e_{l}$ and $d_{n}$.

Let us denote

$\displaystyle M:=\max_{\begin{subarray}{c}j\in W_{1}\\\ (1,2)\in W_{\epsilon
2}\\\ \epsilon\in\\{\pm
1\\}\end{subarray}}\Bigg{\\{}\frac{k_{j}(k_{j}+1)}{2},k_{1}k_{2}\Bigg{\\}},\quad\beta:=\max_{\begin{subarray}{c}j\in
W_{1}\\\ (1,2)\in W_{\epsilon 2}\\\ \epsilon\in\\{\pm
1\\}\end{subarray}}\Bigg{\\{}|1-e^{2i\theta_{j}}|,|1-e^{i(\theta_{1}\pm\theta_{2})}|\Bigg{\\}},$

and
$2w:=\displaystyle\max_{\begin{subarray}{c}j\end{subarray}}\,\\{\,|W_{j}|\,\\}$.
We can write $b_{n}$ as

$\displaystyle b_{n}=(-1)^{n}\sum_{\begin{subarray}{c}\sum n_{j}+m_{12}=n\\\
n_{j},m_{12}\geq 0\\\ j\in W_{1},(1,2)\in W_{\epsilon 2}\\\ \epsilon\in\\{\pm
1\\}\end{subarray}}\prod_{\begin{subarray}{c}j\in W_{1}\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\binom{\frac{k_{j}(k_{j}+1)}{2}+n_{j}}{\frac{k_{j}(k_{j}+1)}{2}}(e^{-2i\epsilon_{j}\theta_{j}}-1)^{n_{j}}$
$\displaystyle\times\prod_{\begin{subarray}{c}(1,2)\in W_{\epsilon 2}\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\binom{k_{1}k_{2}+m_{12}}{k_{1}k_{2}}(e^{-i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})}-1)^{m_{12}}.$

Note that the number of terms such that $\sum n_{j}+m_{12}=n$ with
$n_{j},m_{12}\geq 0$ and $j\in W_{1},\,(1,2)\in W_{\epsilon 2}$ is
$\binom{w+n-1}{w-1}$. Therefore, for large $g$ and $n\geq 1$, we obtain

(4.9)
$\displaystyle|b_{n}|\leq\binom{w+n-1}{w-1}{\binom{M+n}{M}}^{w}{\beta}^{n}\leq
a_{0}n^{t}{\beta}^{n},$

where $a_{0}$, $t$ are constants depend on $w$ and $M$.

Let $r$ be the radius of convergence of $\widetilde{E}(u)$. Note that
$\frac{1}{g}=o(r)$. Hence
$\displaystyle\lim_{n\to\infty}\frac{e_{n+1}}{e_{n}}=\frac{1}{r}=o(g)$, and
this gives

(4.10) $\displaystyle|e_{n}|\leq e_{0}\,a_{1}\,\left(\frac{2}{r}\right)^{n}.$

where $a_{1}\in\mathbb{R}$ depends on $\widetilde{E}$.

Note that,

$\displaystyle\frac{e_{l}d_{V+n-l}}{(V+n-l)!}=\frac{e_{l}}{(V+n-l)!}+\left(\frac{s^{(V+n-l)}_{V+n-l-1}e_{l-1}}{(V+n-l+1)!}+\frac{s^{(V+n-l+1)}_{V+n-l-1}e_{l-2}}{(V+n-l+2)!}+\ldots+\right.$
$\displaystyle\left.\frac{s^{(V+n-2)}_{V+n-l-1}e_{1}}{(V+n-1)!}+\frac{s^{(V+n-1)}_{V+n-l-1}e_{0}}{(V+n)!}\right),$

which implies together with (4.10),

$\frac{e_{l}d_{V+n-l}}{(V+n-l)!}\leq e_{l}+e_{l-1}+\ldots+e_{1}+e_{0}\leq
e_{0}a_{1}\sum_{k=1}^{l}\Big{(}\frac{2}{r}\Big{)}^{k}\ll
l\Big{(}\frac{2}{r}\Big{)}^{l}.$

By using the above bounds, the fact that $b_{0}=1$ and $\frac{1}{r}=o(g)$, the
second sum of (4.8) is bounded by

$\displaystyle\ll_{\bm{k}}\sum_{l=1}^{V}l\left(\frac{2}{r}\right)^{l}\Big{(}\frac{g}{2}\Big{)}^{V-l}\left(1+\sum_{n=1}^{\infty}\frac{n^{t}(g\beta)^{n}}{2^{n}}\right)$
$\displaystyle=o\left({g}^{V}\prod_{j\in{W}^{c}_{1}}\frac{1}{|\theta_{j}|^{k_{j}(k_{j}+1)}}\prod_{\begin{subarray}{c}(1,2)\in{W}^{c}_{\epsilon
2}\\\ \epsilon\in\\{\pm
1\\}\end{subarray}}\frac{1}{|\theta_{1}+\epsilon\theta_{2}|^{2k_{1}k_{2}}}\right).$

Since $|g\beta|<\widetilde{c}<1$ as $g\to\infty$, the inside $n$-sum in the
above expression is $O(1)$. For any $l\geq 1$, using (4.9), we get

$\displaystyle\sum_{n=0}^{\infty}\frac{b_{n+l}d_{n}g^{n}}{2^{n}n!}\ll_{\bm{k}}l^{t}\beta^{l}\sum_{n=0}^{\infty}\frac{(n+l)^{t}}{l^{t}}\frac{(g\beta)^{n}}{2^{n}}\ll_{\bm{k}}l^{t}\beta^{l}\sum_{n=0}^{\infty}\frac{(n+1)^{t}(g\beta)^{n}}{2^{n}}\ll_{\bm{k}}l^{t}\beta^{l}.$

From the fact $\beta/r=o(1)$ and (4.10), the third sum of the equation (4.8)
is bounded above by

$\displaystyle\ll_{\bm{k}}\frac{1}{r^{V}}\sum_{l=1}^{\infty}\left(\frac{2\beta}{r}\right)^{l}l^{t}$
$\displaystyle=o\left(g^{V}\prod_{j\in{W}^{c}_{1}}\frac{1}{|\theta_{j}|^{k_{j}(k_{j}+1)}}\prod_{\begin{subarray}{c}(1,2)\in{W}^{c}_{\epsilon
2}\\\ \epsilon\in\\{\pm
1\\}\end{subarray}}\frac{1}{|\theta_{1}+\epsilon\theta_{2}|^{2k_{1}k_{2}}}\right).$

Finally, we consider the first sum of the equation (4.8). Using the bound of
$b_{n}$ (see (4.9)), we note that

$\displaystyle
1\ll\sum_{n=0}^{\infty}\frac{b_{n}g^{n}}{2^{n}(V+n)!}\ll\frac{1}{V!}+\sum_{n=1}^{\infty}\frac{n^{t}(\beta\,g)^{n}}{2^{n}(V+n)!}=O(1)$

where the implied constant depends on $\widetilde{c}$ and $\bm{k}\,(i.e.,V)$.
Therefore, we conclude that

$\displaystyle\int_{\Gamma}D(u)\,du\sim_{\bm{k},\widetilde{c}}g^{V}\prod_{j\in{W}^{c}_{1}}\frac{1}{|\theta_{j}|^{k_{j}(k_{j}+1)}}\prod_{\begin{subarray}{c}(1,2)\in{W}^{c}_{\epsilon
2}\\\ \epsilon\in\\{\pm
1\\}\end{subarray}}\frac{1}{|\theta_{1}+\epsilon\theta_{2}|^{2k_{1}k_{2}}}$
$\displaystyle\sim_{\bm{k},\widetilde{c}}g^{k_{1}^{2}+k_{2}^{2}}\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{|2\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\frac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\left(\min\Big{\\{}\frac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}},$

as required.

#### Evaluation of infinite “single shift” and “pair shift”

We claim that

(4.11) $\displaystyle\sum_{\begin{subarray}{c}j\in W_{1}\\\
\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\underset{u=\frac{1}{qe^{2i\epsilon_{j}\theta_{j}}}}{\mathrm{Res}}D(u)+\sum_{\begin{subarray}{c}(1,2)\in
W_{\epsilon 2}\\\ \epsilon,\epsilon_{j}\in\\{\pm
1\\}\end{subarray}}\underset{u=\frac{1}{qe^{i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2}})}}{\mathrm{Res}}D(u)\hskip
156.49014pt$
$\displaystyle=o\left(g^{k_{1}^{2}\,+\,k_{2}^{2}}\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{|2\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\frac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\left(\min\Big{\\{}\frac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\right).$

For the sake of simplicity, we will provide all the details of the proof of
the claim (4.11) in the Appendix section.

###  4.2. Estimation of the sum $S_{1}$

We define

$\widetilde{\mathcal{L}}(u,\chi_{D}):={\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}=\sum_{f\in\mathcal{M}}a_{f}\chi_{D}(f)\Big{(}\frac{u}{\sqrt{q}}\Big{)}^{d(f)},$

where $a_{f}$ is defined by (4.2). We begin with the integral

$\displaystyle I=\frac{1}{2\pi
i}\underset{|u|=r}{\oint}\widetilde{\mathcal{L}}(u,\chi_{D})\frac{u^{-(k_{1}+k_{2})X}}{(1-u)}\frac{du}{u},\quad
r<1.$

Integrating term by term to get

$\displaystyle
I=\sum_{f\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}a_{f}\frac{\chi_{D}(f)}{|f|^{1/2}}.$

On the other hand we move the contour of integration to $|u|=q^{y}$,
encountering a simple pole at $u=1$, $y>\frac{1}{2}$. In doing so, we obtain

$\displaystyle I=\widetilde{\mathcal{L}}(1,\chi_{D})+\frac{1}{2\pi
i}\underset{|u|=q^{y}}{\oint}\widetilde{\mathcal{L}}(u,\chi_{D})\frac{u^{-(k_{1}+k_{2})X}}{(1-u)}\frac{du}{u}.$

We use the Lindelöf bound $\widetilde{\mathcal{L}}(u,\chi_{D})\ll
q^{\varepsilon n}$ [[4], Theorem $3.3$]777One can use the Theorem 1.5 to get
the better bound for the integral (4.12), but for our case Lindelöf bound is
enough.  to obtain

(4.12) $\displaystyle\frac{1}{2\pi
i}\underset{|u|=q^{y}}{\oint}\widetilde{\mathcal{L}}(u,\chi_{D})\frac{u^{-(k_{1}+k_{2})X}}{(1-u)}\frac{du}{u}\ll\frac{q^{\varepsilon
n}}{q^{\left((k_{1}+k_{2})X+1\right)y}}.$

It follows that

(4.13)
$\displaystyle\widetilde{\mathcal{L}}(1,\chi_{D})=\sum_{f\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}a_{f}\frac{\chi_{D}(f)}{|f|^{1/2}}+O_{\varepsilon}\left(\frac{q^{\varepsilon
n}}{q^{\left((k_{1}+k_{2})X+1\right)y}}\right).$

From the approximation (4.13),

$\displaystyle S_{1}$
$\displaystyle\gg\Bigg{|}\sum_{D\in\mathcal{H}_{n}}{\mathcal{L}\Big{(}\frac{v_{1}}{q^{1/2+\alpha_{1}}},\chi_{D}\Big{)}}^{k_{1}}{\mathcal{L}\Big{(}\frac{v_{2}}{q^{1/2+\alpha_{2}}},\chi_{D}\Big{)}}^{k_{2}}\overline{\mathcal{L}_{\leq
X}\big{(}\bm{v},\chi_{D}\big{)}}\Bigg{|}\hskip 156.49014pt$
$\displaystyle=\Bigg{|}\sum_{f\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\sum_{f^{\prime}\in\mathcal{M}_{\leq(k_{1}+k_{2})X}}\frac{a_{f}\,\overline{a_{f^{\prime}}}}{|ff^{\prime}|^{1/2}}\sum_{D\in\mathcal{H}_{n}}\chi_{D}(ff^{\prime})\Bigg{|}\,+\,O_{\varepsilon}\left(q^{n\varepsilon}|\mathcal{H}_{n}|\frac{q^{(1/2+\,\varepsilon)(k_{1}+k_{2})X}}{q^{\left((k_{1}+k_{2})X+1\right)y}}\right)$
$\displaystyle=|S_{2}|+O_{\varepsilon}\left(q^{n\varepsilon}\,|\mathcal{H}_{n}|\frac{q^{(1/2\,+\varepsilon)(k_{1}+k_{2})X}}{q^{\left((k_{1}+k_{2})X+1\right)y}}\right).$

We choose $X=\frac{g}{2(k_{1}+k_{2})}$ and $y=\frac{2}{3}$. Hence the estimate
of $S_{2}$ gives us

$\displaystyle
S_{1}\gg_{\bm{k}}\;|\mathcal{H}_{n}|\;g^{k_{1}^{2}+k_{2}^{2}}\,\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{|2\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\frac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}$
$\displaystyle\times\left(\min\Big{\\{}\frac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}+O\left(|\mathcal{H}_{n}|q^{-\frac{g}{12}\,+\,\varepsilon
n}\right).\hskip 14.22636pt$

Inserting the estimates of $S_{1}$ and $S_{2}$ in (4.3) finishes the proof of
Theorem 1.3.

##  5\. Proof of Theorem 1.5

To keep things simple we use the notation $\bm{v}$ instead of $\bm{v}^{(m)}$.
The proof of the Theorem 1.5 will rely on getting an upper bound of the set

$\Upsilon_{n}(\bm{v},V)=\\#\left\\{D\in\mathcal{H}_{n}:\displaystyle\sum_{j=1}^{m}2k_{j}\log\Big{|}\mathcal{L}\Big{(}\frac{v_{j}}{q^{\frac{1}{2}+\alpha_{j}}},\chi_{D}\Big{)}\Big{|}\geq\mu(\bm{v},g)+V\right\\},$

for sufficiently large $n$ and for all $V>2$, where $\mu(\bm{v},g)$ is defined
by (1.5). Recall that

$2g=n-1-\lambda,$

where $g$ and $\lambda$ are defined by (2.5) and (2.4) respectively. We can
write

(5.1)
$\displaystyle\sum_{D\in\mathcal{H}_{n}}\bigg{|}\mathcal{L}\Big{(}\frac{v_{1}}{q^{\frac{1}{2}+\alpha_{1}}},\chi_{D}\Big{)}\bigg{|}^{2k_{1}}\ldots\bigg{|}\mathcal{L}\Big{(}\frac{v_{m}}{q^{\frac{1}{2}+\alpha_{m}}},\chi_{D}\Big{)}\bigg{|}^{2k_{m}}=\displaystyle\int_{-\infty}^{\infty}\Upsilon_{n}(\bm{v},V)\exp{\big{(}\mu(\bm{v},g)+V\big{)}}dV.$

We will estimate an upper bound of $\Upsilon_{n}(\bm{v},V)$ for different
ranges of $V$. The Lemma 3.4 lead us

$\displaystyle\sum_{j=1}^{m}2k_{j}\log\Big{|}\mathcal{L}\Big{(}\frac{v_{j}}{q^{\frac{1}{2}+\alpha_{j}}},\chi_{D}\Big{)}\Big{|}\leq\frac{4g}{N+1}\sum_{j=1}^{m}k_{j}\log\left(\frac{1+q^{-\alpha_{j}(N+1)}}{1+q^{-2(N+1)}}\right)\hskip
142.26378pt$ $\displaystyle+2\Re\sum_{d(f)\leq
N}\sum_{j=1}^{m}k_{j}\frac{a_{\alpha_{j}}\left(d(f)\right)\chi_{D}(f)\Lambda(f){v_{j}}^{d(f)}}{|f|^{\frac{1}{2}}}+O\left(\sum_{j=1}^{m}k_{j}\right)$
$\displaystyle\;\;\leq\frac{4gK}{N+1}\log{2}+2\Re\sum_{d(f)\leq
N}\sum_{j=1}^{m}k_{j}\frac{a_{\alpha_{j}}\left(d(f)\right)\chi_{D}(f)\Lambda(f){v_{j}}^{d(f)}}{|f|^{\frac{1}{2}}}+O(K),$

where

$K=\displaystyle\sum_{j=1}^{m}k_{j}\quad\text{and}\quad
a_{\alpha_{j}}\left(d(f)\right)=\frac{1}{d(f)|f|^{\alpha_{j}}}-\frac{1}{d(f)|f|^{2}}+O\left(\frac{1}{(N+1)q^{(N+1)\alpha_{j}}}\right).$

Applying the prime polynomial theorem, the contribution from square
polynomials $f=P^{2}$ to the second term of the right hand side of the above
inequality is

$\displaystyle
2\Re\sum_{d(P)\leq\frac{N}{2}}\sum_{j=1}^{m}k_{j}\frac{a_{\alpha_{j}}\left(2d(P)\right)\chi_{D}(P)d(P){v_{j}}^{2d(P)}}{|P|}+O(\log\log
n)$ $\displaystyle\leq\mu(\bm{v},g)+\frac{2gK}{N+1}+O(\log\log n),$

where the error term $O(\log\log n)$ comes from the sum over $P$ such that
$P|D$. Also it is easy to verify that the contribution from $f=P^{r}$ with
$r\geq 3$ is $O(1)$. Therefore, we deduce that

$\displaystyle\sum_{j=1}^{m}2k_{j}\log\Big{|}\mathcal{L}\Big{(}\frac{v_{j}}{q^{\frac{1}{2}+\alpha_{j}}},\chi_{D}\Big{)}\Big{|}\leq
S_{1}(D)+S_{2}(D)+\mu(\bm{v},g)+\frac{5gK}{N+1}+O(\log\log n),$

where

$\displaystyle S_{1}(D)=2\displaystyle\sum_{d(P)\leq
N_{0}}\frac{\chi_{D}(P)}{|P|^{1/2}}\displaystyle\sum_{j=1}^{m}k_{j}a_{\alpha_{j}}\left(d(P)\right)d(P)\cos(\theta_{j}d(P)),$
$\displaystyle S_{2}(D)=2\displaystyle\sum_{N_{0}<d(P)\leq
N}\frac{\chi_{D}(P)}{|P|^{1/2}}\displaystyle\sum_{j=1}^{m}k_{j}a_{\alpha_{j}}\left(d(P)\right)d(P)\cos(\theta_{j}d(P)).$

We rewrite $\sigma(\bm{v},g)$ as

$\sigma(\bm{v},g)=2\left(\sum_{j=1}^{m}k_{j}^{2}\right)\log{g}+2\sum_{j=1}^{m}k_{j}^{2}F_{j}+4\displaystyle\sum_{i<j}k_{i}k_{j}F_{i,j},$

where

$F_{j}=\log\left(\min\Big{\\{}\frac{1}{2|\theta_{j}|},g\Big{\\}}\right)\,\text{
and
}\,F_{i,j}=\log\left(\min\Big{\\{}\tfrac{1}{|\theta_{i}-\theta_{j}|},g\Big{\\}}\right)+\log\left(\min\Big{\\{}\tfrac{1}{|\theta_{i}+\theta_{j}|},g\Big{\\}}\right).$

From now onward, for the sake of simplicity we write $\sigma(\bm{v},g)$ simply
as $\bm{\sigma}$. We consider various different range of $V$. The range
$-\infty<V\leq\sqrt{\log{g}}$ yields

$\displaystyle\int_{-\infty}^{\infty}\Upsilon_{n}(\bm{v},V)\exp{\big{(}\mu(\bm{v},g)+V\big{)}}dV\ll|\mathcal{H}_{n}|\exp{\big{(}\sqrt{\log{g}}+\mu(\bm{v},g\big{)}}\ll|\mathcal{H}_{n}|g^{o(1)}\exp{(\mu(\bm{v},g))}.$

Applying Lemma 3.1, it is enough to assume that $\sqrt{\log{g}}\leq
V\leq\frac{Kg}{\log_{q}g}$. We define the quantity $A$ by

$\displaystyle
A=\left\\{\begin{array}[c]{ll}\frac{\log\bm{\sigma}}{2},&\,\text{if\,}\sqrt{\log{g}}\leq
V\leq\bm{\sigma},\vspace{1mm}\\\
\frac{\bm{\sigma}\log\bm{\sigma}}{2V},&\,\text{if\,}\bm{\sigma}\leq
V\leq\frac{\bm{\sigma}\log\bm{\sigma}}{25K},\vspace{1mm}\\\
7K,&\,\text{if\,}V>\frac{\bm{\sigma}\log\bm{\sigma}}{25K}.\end{array}\right.$

Let us consider

$\frac{g}{N+1}=\frac{V}{A}\quad\text{ and }\quad N_{0}=\frac{N}{\log_{q}g}.$

Notice that, if $D\in\Upsilon_{n}(\bm{v},V)$ then we must have either

$S_{1}(D)\geq V(1-\frac{6K}{A}):=V_{1}\quad\text{ or }\quad
S_{2}(D)\geq\frac{KV}{A}:=V_{2}.$

To determine an upper bound of $\Upsilon_{n}(\bm{v},V)$, we will actually
examine the set

$\Upsilon_{n}(\bm{v},V_{i})=\\#\left\\{D\in\mathcal{H}_{n}:S_{i}(D)\geq
V_{i}\right\\},$

for $i=1,2$. We set
$a_{j}(P):=a_{\alpha_{j}}\left(d(P)\right)d(P)\cos(\theta_{j}d(P))$. So

$a_{j}(P)=\frac{\cos(\theta_{j}d(P))}{|P|^{\alpha_{j}}}-\frac{\cos(\theta_{j}d(P))}{|P|^{2}}+O\left(\frac{d(P)}{(N+1)q^{(N+1)\alpha_{j}}}\right)\ll
1.$

Using Lemma 3.6, we obtain

$\displaystyle\sum_{D\in\mathcal{H}_{n}}|S_{2}(D)|^{2l}$
$\displaystyle\ll|\mathcal{H}_{n}|\frac{(2l)!}{l!\,2^{l}}\Bigg{(}\sum_{N_{0}<d(P)\leq
N}\frac{(\sum_{j=1}^{m}2k_{j})^{2}}{|P|}\Bigg{)}^{l}$
$\displaystyle\ll|\mathcal{H}_{n}|\frac{(2l)!}{l!2^{l}}\left(4K^{2}\left(\log\log_{q}g+O(1)\right)\right)^{l},$

for any $l$ such that $2lN\leq n$, which implies that
$l\leq\frac{g}{N}+\frac{1}{2N}\leq\frac{2V}{A}.$

Therefore, by using Markov’s inequality and Stirling’s formula, it follows
that

$\displaystyle\Upsilon_{n}(\bm{v},V_{2})$
$\displaystyle\leq{V_{2}}^{-2l}\Big{(}\sum_{D\in\mathcal{H}_{n}}|S_{2}(D)|^{2l}\Big{)}$
$\displaystyle\ll|\mathcal{H}_{n}|\Big{(}\frac{A}{KV}\Big{)}^{2l}\frac{(2l)!}{l!2^{l}}\left(4K^{2}\left(\log\log_{q}g+O(1)\right)\right)^{l}$
$\displaystyle\ll|\mathcal{H}_{n}|\exp(-\frac{V}{2A}\log V).$

Again applying Lemma 3.6 and Stirling’s formula, we get

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\big{|}S_{1}(D)\big{|}^{2l}\ll|\mathcal{H}_{n}|\frac{(2l)!}{l!2^{l}}\Bigg{(}\sum_{d(P)\leq
N_{0}}\frac{1}{|P|}\bigg{(}\sum_{j=1}^{m}\frac{2k_{j}\cos(\theta_{j}d(P))}{|P|^{\alpha_{j}}}\bigg{)}^{2}\Bigg{)}^{l}\hskip
142.26378pt$
$\displaystyle\ll|\mathcal{H}_{n}|\frac{(2l)!}{l!2^{l}}\Bigg{(}\sum_{d(P)\leq
N_{0}}4\bigg{(}\sum_{j=1}^{m}\frac{{k_{j}}^{2}\cos^{2}(\theta_{j}d(P))}{|P|^{1+2\alpha_{j}}}+2\sum_{i<j}k_{i}k_{j}\frac{\cos(\theta_{i}d(P))\cos(\theta_{j}d(P))}{|P|^{1+(\alpha_{i}+\alpha_{j})}}\bigg{)}\Bigg{)}^{l}$
$\displaystyle\ll|\mathcal{H}_{n}|\left(\frac{l\bm{\sigma}}{e}\right)^{l},\hskip
352.814pt$

for any $l$ such that $2lN_{0}\leq n$, which implies that
$l\leq\frac{V}{A}\log_{q}g$. Markov’s inequality gives us

$\displaystyle\Upsilon_{n}(\bm{v},V_{1})\ll{V_{1}}^{-2l}\left(\sum_{D\in\mathcal{H}_{n}}|S_{1}(D)|^{2l}\right)\ll|\mathcal{H}_{n}|\left(\frac{l\bm{\sigma}}{e{V_{1}}^{2}}\right)^{l}.$

It is now convenient to consider the case when
$V\leq\frac{\bm{\sigma}^{2}}{K^{3}}$ and the case
$V>\frac{\bm{\sigma}^{2}}{K^{3}}$ separately.

${Case\,1.}\;$ Assume that $V\leq\frac{\bm{\sigma}^{2}}{K^{3}}$. We choose
$l=\lfloor\frac{{V_{1}}^{2}}{\bm{\sigma}}\rfloor$. The definition of $A$ and
this choice of $l$ implies that $l\leq\frac{V}{A}\log_{q}g$. In this case, we
find that

$\displaystyle\Upsilon_{n}(\bm{v},V_{1})\ll|\mathcal{H}_{n}|\,\exp\left(l\log\left(\frac{l\bm{\sigma}}{e{V_{1}}^{2}}\right)\right)\ll|\mathcal{H}_{n}|\,\exp\left(-\frac{{V_{1}}^{2}}{\bm{\sigma}}\right).$

${Case\,2.}\;$ Assume that $V>\frac{\bm{\sigma}^{2}}{K^{3}}$. We choose
$l=\lfloor 10V\rfloor$. Again from the definition of $A$, it is easy to see
that this choice $l$ satisfies $l\leq\frac{V}{A}\log_{q}g$. Notice that
$V>\frac{\bm{\sigma}^{2}}{K^{3}}$, implies
$\log{V}>2\log{\bm{\sigma}}-3\log{K}$. So, we have

$A=K\quad\text{ and }\quad{V_{1}}^{2}=25{V}^{2}.$

Hence, we conclude that

$\displaystyle\Upsilon_{n}(\bm{v},V_{1})$
$\displaystyle\ll|\mathcal{H}_{n}|\exp\left(10V\log\left(\frac{10V\bm{\sigma}}{e{V_{1}}^{2}}\right)\right)$
$\displaystyle\ll|\mathcal{H}_{n}|\exp\left(-4V\log{V}\right),$

for sufficiently large $g$.

Therefore combining the above estimates, we deduce that

(5.2)
$\displaystyle\Upsilon_{n}(\bm{v},V)\ll|\mathcal{H}_{n}|\left\\{\exp\left(-\tfrac{V}{2A}\log
V\right)+\exp\left(-\tfrac{{V_{1}}^{2}}{\bm{\sigma}}\right)+\exp\left(-4V\log{V}\right)\right\\}.$

We extract the value of $V_{1}$ for various range of $V$ comes from the
definition of $A$.

If $\sqrt{\log{g}}\leq V\leq\bm{\sigma}$, then

$A=\frac{1}{2}\log\bm{\sigma}\quad\text{and}\quad
V_{1}=V\Big{(}1-\frac{12K}{\log\bm{\sigma}}\Big{)}.$

So, for sufficiently large $g$, (5.2) implies that

$\displaystyle\Upsilon_{n}(\bm{v},V)$
$\displaystyle\ll|\mathcal{H}_{n}|\exp\left(-\tfrac{V^{2}}{\bm{\sigma}}\left(1-\tfrac{12K}{\log\bm{\sigma}}\right)^{2}\right)$
$\displaystyle\ll|\mathcal{H}_{n}|\exp\left(-\tfrac{V^{2}}{\bm{\sigma}}\left(1-\tfrac{24K}{\log\bm{\sigma}}\right)\right).$

If $\bm{\sigma}\leq V\leq\frac{1}{25K}\bm{\sigma}\log\bm{\sigma}$, then

$A=\frac{\bm{\sigma}\log\bm{\sigma}}{2V}\quad\text{ and }\quad
V_{1}=V\Big{(}1-\tfrac{12KV}{\bm{\sigma}\log\bm{\sigma}}\Big{)}.$

For this range of $V$,
$\frac{\log{V}}{\bm{\sigma}\log\bm{\sigma}}>\frac{1}{\bm{\sigma}}$ and hence
from (5.2) we obtain

$\displaystyle\Upsilon_{n}(\bm{v},V)$
$\displaystyle\ll|\mathcal{H}_{n}|\bigg{\\{}\exp\left(-\frac{V^{2}\log{V}}{\bm{\sigma}\log\bm{\sigma}}\right)+\exp\left(-4V\log{V}\right)\bigg{.}$
$\displaystyle\bigg{.}\hskip
85.35826pt+\exp\left(-\tfrac{V^{2}}{\bm{\sigma}}\left(1-\frac{12KV}{\bm{\sigma}\log\bm{\sigma}}\right)^{2}\right)\bigg{\\}}$
$\displaystyle\ll|\mathcal{H}_{n}|\exp\left(-\frac{V^{2}}{\bm{\sigma}}\left(1-\frac{24KV}{\bm{\sigma}\log\bm{\sigma}}\right)\right).$

Finally, if $V>\frac{1}{25K}\bm{\sigma}\log\bm{\sigma}$, then

$A=7K\quad\text{ and }\quad V_{1}=\frac{V}{7}.$

So from (5.2), we get that

$\displaystyle\Upsilon_{n}(\bm{v},V)\ll|\mathcal{H}_{n}|\exp\left(-\tfrac{V}{98K}\log
V\right).$

Adding these estimates in (5.2) for different range of $V$, we conclude that

(5.5)
$\displaystyle\Upsilon_{n}(\bm{v},V)\ll\left\\{\begin{array}[c]{ll}|\mathcal{H}_{n}|n^{\varepsilon}\,\exp\left(-\frac{V^{2}}{\bm{\sigma}}\right)&,\,\text{if\,}3\leq
V\leq 2021\bm{\sigma},\vspace{1mm}\\\
|\mathcal{H}_{n}|n^{\varepsilon}\exp\left(-4V\right)&,\,\text{if\,}V>2021\bm{\sigma}.\end{array}\right.$

Inserting (5.5) in (5.1) finishes the proof of Theorem 1.5.

##  6\. Proof of Theorem 1.6

Let $C_{1/g}$ be the circle in the complex plane whose center is origin and
radius is $\frac{1}{g}$. By Cauchy’s integral formula

$\mathcal{L}^{(l)}(q^{-1/2},\chi_{D})=\frac{l!}{2\pi
i}\oint_{C_{1/g}}L\left(\frac{1}{2}+\theta,\chi_{D}\right)\,\frac{d\theta}{{\theta}^{l+1}},$

Notice that if $\theta=\alpha-\frac{it}{\log q}$, then

$L\Big{(}\frac{1}{2}+\theta,\chi_{D}\Big{)}=\mathcal{L}\Big{(}\frac{v}{q^{\alpha+\,1/2}},\chi_{D}\Big{)},$

where $\alpha=O\left(\frac{1}{g}\right)$. Therefore, applying Hölder’s
inequality, we see that

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\big{|}\mathcal{L}^{(l)}(q^{-1/2},\chi_{D})\big{|}^{k}$
$\displaystyle\leq\left(\frac{l!}{2\pi}\right)^{k}\Big{(}\sum_{D\in\mathcal{H}_{n}}\oint_{C_{1/g}}\big{|}L\Big{(}\frac{1}{2}+\theta,\chi_{D}\Big{)}\big{|}^{k}\,|d\theta|\Big{)}\Big{(}\oint_{C_{1/g}}|\theta|^{-\frac{k(l+1)}{(k-1)}}|d\theta|\Big{)}^{(k-1)}$
$\displaystyle\ll\left(\frac{l!}{2\pi}\right)^{k}\left(\frac{2\pi}{g}\right)^{k-1}\left(\frac{g}{2\pi}\right)^{k(l+1)}\Big{(}\sum_{D\in\mathcal{H}_{n}}\oint_{C_{1/g}}\big{|}L\Big{(}\frac{1}{2}+\theta,\chi_{D}\Big{)}\big{|}^{k}\,|d\theta|\Big{)}$
$\displaystyle\ll\left(\frac{l!}{2\pi}\right)^{k}\left(\frac{2\pi}{g}\right)^{k}\left(\frac{g}{2\pi}\right)^{k(l+1)}\underset{|\theta|\leq\frac{1}{g}}{\max}\sum_{D\in\mathcal{H}_{n}}\big{|}L\Big{(}\frac{1}{2}+\theta,\chi_{D}\Big{)}\big{|}^{k}.$

As a direct application of Theorem 1.5, we obtain

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\Big{|}\mathcal{L}\Big{(}\frac{v}{q^{\alpha+\,1/2}},\chi_{D}\Big{)}\Big{|}^{k}\ll_{\varepsilon}|\mathcal{H}_{n}|\,{g}^{\frac{k(k+1)}{2}+\varepsilon}.$

Using this upper bound to the above inequality, we conclude that

$\displaystyle\sum_{D\in\mathcal{H}_{n}}\big{|}\mathcal{L}^{(l)}(q^{-1/2},\chi_{D})\big{|}^{k}\ll_{\varepsilon}|\mathcal{H}_{n}|\,{g}^{\frac{k(k+1)}{2}+kl+\varepsilon}.$

Acknowledgements: We thank the anonymous referees for their valuable comments
and insightful suggestions that have improved the quality of the manuscript.

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##  7\. Appendix

###  7.1. Proof of claim (4.11)

We have to show that for $j\in{W}^{c}_{1}$,

(7.1) $\displaystyle\underset{u=\frac{1}{qe^{2i\theta_{j}}}}{Res}D(u)=\hskip
341.43306pt$ $\displaystyle
o\left(g^{k_{1}^{2}\,+\,k_{2}^{2}}\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{2|\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\tfrac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\left(\min\Big{\\{}\tfrac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\right),$

and for $(1,2)\in{W}^{c}_{2\epsilon}$, $\epsilon\in\\{\pm 1\\}$888To estimate
infinite pair shift for Theorem 1.5, one can follow the article of V. Chande
[[7], Appendix].,

(7.2)
$\displaystyle\underset{u=\frac{1}{qe^{i(\theta_{1}+\epsilon\theta_{2})}}}{Res}D(u)=\hskip
312.9803pt$ $\displaystyle
o\left(g^{k_{1}^{2}\,+\,k_{2}^{2}}\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{2|\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\tfrac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\left(\min\Big{\\{}\tfrac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\right).$

We will prove the claim (7.1) and proof of the claim (7.2) follows in the
similar way. We assume that $\epsilon,\epsilon_{j}\in\\{1,-1\\}$ for $j=1,2$.
To prove the claim (7.1), without loss of generality, we assume that $1\in
W^{c}_{1}$, so $(1,2)\in W_{-2}^{c}$. Note that if $2\in{W}^{c}_{1}$ and
$(1,2)\in{W}^{c}_{2}$, then they are not closed to each other i.e.,
$|\theta_{1}-\theta_{2}|\gg\frac{1}{g}$,
$|\theta_{1}-(\theta_{1}-\theta_{2})|=|\theta_{2}|\gg\frac{1}{g}$ and
$|\theta_{2}-(\theta_{1}+\theta_{2})|=|\theta_{1}|\gg\frac{1}{g}$, otherwise
they will contained in the sets $W_{1}$ and $W_{2}$ respectively. By Cauchy’s
theorem, we obtain

$\underset{u=1/qe^{2i\theta_{j}}}{Res}D(u)=\underset{\widetilde{C}}{\oint}D(u)\,du,\quad
j=1,2,$

where $\widetilde{C}$ is the circle centered at $u=1/qe^{2i\theta_{j}}$ with
radius $\frac{\widetilde{c}}{g}$ and $\widetilde{c}$ is defined by (4.5). Note
that, $\frac{1}{g}=o(|\theta_{j}|)$ and $\theta_{j}=o(1)$ for all $j$. For $u$
on the circle $\widetilde{C}$, we write

$\mathcal{Z}(u)=(1-qu)^{-1}=(1-e^{-2i\theta_{1}})^{-1}\left(1+\frac{e^{-2i\theta_{1}}(1-que^{2i\theta_{1}})}{1-e^{-2i\theta_{1}}}\right)^{-1}.$

Therefore, we get

$|\mathcal{Z}(u)|\ll\frac{1}{|\theta_{1}|}.$

If $2\in W_{1}$, then it is easy to see that
$|\theta_{1}\pm\theta_{2}|\sim|\theta_{1}|$. For $u$ on the circle
$\widetilde{C}$,

$\mathcal{Z}(ue^{2i\epsilon_{j}\theta_{2}})=(1-que^{2i\epsilon_{j}\theta_{2}})^{-1}=(1-e^{-2i\theta_{1}})^{-1}\left(1+\frac{e^{-2i\theta_{1}}(1-que^{2i(\epsilon_{j}\theta_{2}-\theta_{1})})}{1-e^{-2i\theta_{1}}}\right)^{-1},$

which implies that

$|\mathcal{Z}(ue^{2i\epsilon_{j}\theta_{2}})|\ll\frac{1}{|\theta_{1}|}.$

Also, if $(1,2)\in W_{2}$, then for $u$ on the circle $\widetilde{C}$, we see
that

$|\mathcal{Z}(ue^{2i(\epsilon_{1}\theta_{1}+\epsilon_{2}\theta_{2})})|\ll\frac{1}{|\theta_{1}|}.$

For elements in the infinite single shift and pair shift, we have to partition
the sets ${W}^{c}_{1}$, ${W}^{c}_{\epsilon 2}$ into three different subsets to
estimate bounds for the corresponding zeta functions. For $2\in{W}^{c}_{1}$,
we divide the set ${W}^{c}_{1}$ into three subsets. First we define

${W}^{c}_{11}:=\left\\{2\in{W}^{c}_{1}:\underset{g\to\infty}{\lim}\,\frac{|\theta_{1}|}{|\theta_{2}|}\,<\,+\infty\;\text{and}\;\underset{g\to\infty}{\lim}\,\frac{\theta_{1}}{\theta_{2}}\neq
1\right\\}.$

If $2\in{W}^{c}_{11}$, then for $u$ on the circle $\widetilde{C}$,

$|\mathcal{Z}(ue^{2i\epsilon_{j}\theta_{2}})|\ll\frac{1}{|\theta_{2}|}.$

Next, we consider

${W}^{c}_{12}=\left\\{2\in{W}^{c}_{1}:\underset{g\to\infty}{\lim}\,\frac{|\theta_{1}|}{|\theta_{2}|}=\,\infty\right\\}.$

For $2\in{W}^{c}_{12}$ and $u$ on the circle $\widetilde{C}$, we obtain

$|\mathcal{Z}(ue^{2i\epsilon_{j}\theta_{2}})|\ll\frac{1}{|\theta_{1}|}.$

Lastly, let

${W}^{c}_{13}=\left\\{2\in{W}^{c}_{1}:\underset{g\to\infty}{\lim}\,\frac{\theta_{1}}{\theta_{2}}=\,1\right\\}.$

For $2\in{W}^{c}_{13}$ and $u$ on the circle $\widetilde{C}$,

$|\mathcal{Z}(ue^{2i\epsilon_{j}\theta_{2}})|\ll\frac{1}{|\theta_{1}-\theta_{2}|}.$

Similarly, for $(1,2)\in{W}^{c}_{\epsilon 2}$, we define

${}^{1}{W}_{\epsilon 2}^{c}=\left\\{(1,2)\in{W}^{c}_{\epsilon
2}\,:\,\underset{g\to\infty}{\lim}\,\frac{|\theta_{1}|}{|\theta_{1}-\epsilon\theta_{2}|}\,<\,+\infty\;\text{and}\;\underset{g\to\infty}{\lim}\,\frac{\theta_{1}}{(\theta_{1}-\epsilon\theta_{2})}\neq
1\right\\}.$

In this case, for $u$ on the circle $\widetilde{C}$,

$|\mathcal{Z}(ue^{i(\theta_{1}-\epsilon\theta_{2})})|\ll\frac{1}{|\theta_{1}-\theta_{2}|}.$

Let

${}^{2}{W}_{\epsilon 2}^{c}=\left\\{(1,2)\in{W}^{c}_{\epsilon
2}:\underset{g\to\infty}{\lim}\,\frac{|\theta_{1}|}{|\theta_{1}-\epsilon\theta_{2}|}\,=\,+\infty\;\right\\}.$

Inside the set ${}^{2}{W}_{\epsilon 2}^{c}$, for $u$ on the circle
$\widetilde{C}$,

$|\mathcal{Z}(ue^{i(\theta_{1}-\epsilon\theta_{2})})|\ll\frac{1}{|\theta_{1}|}.$

Lastly, we consider

${}^{3}{W}_{\epsilon 2}^{c}=\left\\{(1,2)\in{W}^{c}_{\epsilon
2}:\underset{g\to\infty}{\lim}\,\frac{\theta_{1}}{(\theta_{1}-\epsilon\theta_{2})}=1\right\\}.$

For $u$ on the circle $\widetilde{C}$,

$|\mathcal{Z}(ue^{i(\theta_{1}-\epsilon\theta_{2})})|\ll\frac{1}{|\theta_{2}|}.$

Using these bounds for the zeta functions, we conclude that

$\displaystyle\underset{\widetilde{C}}{\oint}D(u)\,du\;\ll\,{g}^{\frac{k_{1}(k_{1}+1)}{2}-1}\,{|\theta_{1}|}^{-\left(k_{1}^{2}\,+\,k_{2}^{2}+\frac{k_{1}(k_{1}+1)}{2}\right)}\,\min\left\\{\frac{1}{|\theta_{1}|},\frac{1}{|\theta_{2}|},\frac{1}{|\theta_{1}-\theta_{2}|}\right\\}^{k_{2}(k_{2}+1)}$
$\displaystyle\times\min\left\\{\frac{1}{|\theta_{1}|},\frac{1}{|\theta_{2}|},\frac{1}{|\theta_{1}-\theta_{2}|}\right\\}^{2k_{1}k_{2}}\min\left\\{\frac{1}{|\theta_{1}|},\frac{1}{|\theta_{2}|},\frac{1}{|\theta_{1}+\theta_{2}|}\right\\}^{2k_{1}k_{2}}$

Using the fact $\frac{1}{|\theta_{j}|}=o(g)$, one can easily cheek that the
integral $\underset{\widetilde{C}}{\oint}D(u)\,du$ is equal to

$o\left(g^{k_{1}^{2}\,+\,k_{2}^{2}}\prod_{j=1}^{2}\left(\min\Big{\\{}\frac{1}{|2\theta_{j}|},g\Big{\\}}\right)^{k_{j}(k_{j}+1)}\left(\min\Big{\\{}\frac{1}{|\theta_{1}-\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\left(\min\Big{\\{}\frac{1}{|\theta_{1}+\theta_{2}|},g\Big{\\}}\right)^{2k_{1}k_{2}}\right),$

and we obtain the claim (7.1).

###  7.2. Deduction of $d_{n}$

We start with the expression (4.7), i.e.,

$\displaystyle
e_{0}b_{n}\frac{F_{V+n}((k_{1}+k_{2})X)}{(V+n)!}\,+\sum_{l=1}^{V+n}e_{l}b_{n}\frac{F_{V+n-l}((k_{1}+k_{2})X)}{(V+n-l)!}$

where $F_{n}(x)=x(x+1)(x+2)\ldots(x+n-1)$, for $n\geq 2$ and
$F_{0}(x)=1,\,F_{1}(x)=x$. We expand $F_{n}(x)$ to get

$\displaystyle
F_{n}(x)=x\left(x^{n-1}+s^{(n-1)}_{n-2}x^{n-2}+s^{(n-1)}_{n-3}x^{n-3}+s^{(n-1)}_{n-4}x^{n-4}+\ldots+s^{(n-1)}_{0}\right)$

with

$s^{(k)}_{k-i}=\sum_{1\leq l_{1}<\ldots<l_{i}\leq k}l_{1}\ldots
l_{i}\,,\;\;i=1,2,\ldots,k.$

This gives us

$\displaystyle
e_{0}b_{n}\frac{F_{V+n}(x)}{(V+n)!}\,+e_{1}b_{n}\frac{F_{V+n-1}(x)}{(V+n-1)!}+e_{2}b_{n}\frac{F_{V+n-2}(x)}{(V+n-2)!}+e_{3}b_{n}\frac{F_{V+n-3}(x)}{(V+n-3)!}$
$\displaystyle+\ldots+e_{V+n-1}b_{n}F_{1}(x)+e_{V+n}b_{n}F_{0}(x)$
$\displaystyle=\frac{e_{0}b_{n}}{(V+n)!}\left(x^{V+n}+s^{(V+n-1)}_{V+n-2}x^{V+n-1}+s^{(V+n-1)}_{V+n-3}x^{V+n-2}+\ldots+s^{(V+n-1)}_{0}x\right)\hskip
14.22636pt$
$\displaystyle+\frac{e_{1}b_{n}}{(V+n-1)!}\left(x^{V+n-1}+s^{(V+n-2)}_{V+n-3}x^{V+n-2}+s^{(V+n-2)}_{V+n-4}x^{V+n-3}+\ldots+s^{(V+n-2)}_{0}x\right)$
$\displaystyle+\frac{e_{2}b_{n}}{(V+n-2)!}\left(x^{V+n-2}+s^{(V+n-3)}_{V+n-4}x^{V+n-3}+s^{(V+n-3)}_{V+n-5}x^{V+n-4}+\ldots+s^{(V+n-3)}_{0}x\right)$
$\displaystyle+\ldots+\hskip 227.62204pt$
$\displaystyle+\frac{e_{V+n-3}b_{n}}{3!}\left(x^{3}+s^{(2)}_{1}x^{2}+s^{(2)}_{0}x\right)+\frac{e_{V+n-2}b_{n}}{2!}\left(x^{2}+s^{(1)}_{0}x\right)+e_{V+n-1}b_{n}x+e_{V+n}b_{n}.$

$\displaystyle:=\frac{e_{0}b_{n}}{(V+n)!}x^{V+n}+\frac{e_{1}b_{n}d_{V+n-1}}{(V+n-1)!}x^{V+n-1}+\frac{e_{2}b_{n}d_{V+n-2}}{(V+n-2)!}x^{V+n-2}+\frac{e_{3}b_{n}d_{V+n-3}}{(V+n-3)!}x^{V+n-3}$
$\displaystyle+\ldots+\frac{e_{V+n-3}b_{n}\,d_{3}}{3!}x^{3}+\frac{e_{V+n-2}b_{n}\,d_{2}}{2!}x^{2}+e_{V+n-1}b_{n}\,d_{1}x+e_{V+n}b_{n}\,d_{0},$

where $d_{0}=1$, and $1\leq l\leq V+n-1$,

$d_{l}=1+\frac{l!}{e_{V+n-l}}\left(\frac{s^{(l)}_{l-1}e_{V+n-(l+1)}}{(l+1)!}+\frac{s^{(l+1)}_{l-1}e_{V+n-(l+2)}}{(l+2)!}+\ldots+\frac{s^{(V+n-2)}_{l-1}e_{1}}{(V+n-1)!}+\frac{s^{(V+n-1)}_{l-1}e_{0}}{(V+n)!}\right).$