# Twists by Dirichlet characters and polynomial Euler products of
$L$-functions, II

J.Kaczorowski and A.Perelli

In memory of Eduard Wirsing

Abstract. In a previous paper we proved that if an $L$-function $F$ from the
Selberg class has degree $2$, its conductor $q_{F}$ is a prime number and $F$
is weakly twist-regular at all primes $p\neq q_{F}$, then $F$ has a polynomial
Euler product. In this paper we extend this result to $L$-functions of degree
2 with square-free conductor $q_{F}$, which are weakly twist-regular at all
primes $p\nmid q_{F}$.

Mathematics Subject Classification (2010): 11M41

Keywords: Twists by Dirichlet characters; Euler products; Selberg class

##  1\. Introduction

In [5] we proved, among other things, that if an $L$-function $F$ from the
Selberg class ${\mathcal{S}}$ has degree $2$, its conductor $q_{F}$ is a prime
number, and $F$ is weakly twist-regular at all primes $p\neq q_{F}$, then $F$
has a polynomial Euler product. In this paper we keep the notation from [5]
but, for the reader’s convenience, we recall some basic definitions in Section
2. The aim of this paper is to extend the above result as follows.

Theorem 1. Let $F\in{\mathcal{S}}$ be of degree $2$ and its conductor $q_{F}$
be square-free. If $F$ is weakly twist-regular at all primes $p\nmid q_{F}$,
then $F$ has a polynomial Euler product.

This is not a straightforward generalization; indeed, apart from the use of
Theorem 2 below, a non-trivial extension of the method in [5] is necessary. On
the other hand, the present method cannot settle the problem in full
generality, i.e. for all integer conductors, and some new ideas will probably
be needed to prove the full result.

One of the main tools in the proof is the following transformation formula for
linear twists of $L$-functions from the extended Selberg class, which is of
independent interest.

Theorem 2. Let $F\in{\mathcal{S}}^{\sharp}$ with $d_{F}=2$, and let
$\alpha>0$. Then for every integer $K>0$ there exist polynomials
$Q_{0}(s),...,Q_{K}(s)$, with $Q_{0}(s)\equiv 1$, such that

$F(s,\alpha)=-i\omega_{F}^{*}(\sqrt{q_{F}}\alpha)^{2s-1+2i\theta_{F}}\sum_{\nu=0}^{K}\big{(}\frac{iq_{F}\alpha}{2\pi}\big{)}^{\nu}Q_{\nu}(s)\overline{F}\big{(}s+\nu+2i\theta_{F},-\frac{1}{q_{F}\alpha}\big{)}+H_{K}(s,\alpha).$
(1.1)

Here $H_{K}(s,\alpha)$ is holomorphic for $-K+\frac{1}{2}<\sigma<2$ and
satisfies

$H_{K}(s,\alpha)\ll(|s|+1)^{2K+A}$ (1.2)

with a certain constant $A=A(F,\alpha)>0$. Moreover, $\deg Q_{\nu}=2\nu$ and

$Q_{\nu}(s)\ll\frac{(A(|s|+1))^{2\nu}}{\nu!}\hskip 42.67912pt\text{for}\
1\leq\nu\leq\min(|s|,K).$ (1.3)

This should be compared with Theorem 1.2 in [4]. Apart from the value
$2\theta_{F}$ in place of $\theta_{F}$ in (1.1), due to a slight change in the
definition of $\theta_{F}$ compared with [4] (see next section), the main
difference is in the estimate for the size of $H_{K}(s,\alpha)$ and in the
range for $s$ in which it holds. Precisely, Theorem 1.2 in [4] states that

$H_{K}(s,\alpha)\ll(AK)^{K}\hskip 42.67912pt\text{for
$-K+\frac{1}{2}<\sigma<2$, $|s|\leq 2K$}.$ (1.4)

In the proof of Theorem 1, we consider shifts of $L$-functions of the form
$F(s+i\tau)$ with $\tau\to\infty$ and hence reasonable control upon the size
of $H_{K}(s,\alpha)$ is needed in half-planes rather than in discs. If fact,
for $\tau\to\infty$, (1.4) gives an estimate which is far too weak for the
proof of Theorem 1. In contrast, (1.2) secures a polynomial growth which is
exactly what is needed. In principle, the main structure of the proof of
Theorem 2 is the same as that of Theorem 1.2 in [4]. Nevertheless, the new
situation where $s$ is in a half-plane forces the introduction of significant
technical changes in the proof.

Acknowledgements. We wish to warmly thank the referee for carefully reading
our manuscript and for pointing out many inaccuracies. This research was
partially supported by the Istituto Nazionale di Alta Matematica, by the MIUR
grant PRIN-2017 “Geometric, algebraic and analytic methods in arithmetic” and
by grant 2021/41/BST1/00241 “Analytic methods in number theory” from the
National Science Centre, Poland.

##  2\. Definitions

Throughout the paper we write $s=\sigma+it$, $e(x)=e^{2\pi ix}$ and
$\overline{f}(s)$ for $\overline{f(\overline{s})}$. The extended Selberg class
${\mathcal{S}}^{\sharp}$ consists of non-identically vanishing Dirichlet
series

$F(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}$

absolutely convergent for $\sigma>1$, such that $(s-1)^{m}F(s)$ is entire of
finite order for some integer $m\geq 0$, and satisfying a functional equation
of type

$F(s)\gamma(s)=\omega\overline{\gamma}(1-s)\overline{F}(1-s),$

where $|\omega|=1$ and the $\gamma$-factor

$\gamma(s)=Q^{s}\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})$

has $Q>0$, $r\geq 0$, $\lambda_{j}>0$ and $\Re(\mu_{j})\geq 0$. Note that the
conjugate function $\overline{F}$ has conjugated coefficients
$\overline{a(n)}$. The Selberg class ${\mathcal{S}}$ is the subclass of
${\mathcal{S}}^{\sharp}$ of the functions satisfying the Ramanujan conjecture
$a(n)\ll n^{\varepsilon}$ and with an Euler product of the form

$F(s)=\prod_{p}F_{p}(s),\quad\text{where}\quad
F_{p}(s)=\sum_{k=0}^{\infty}\frac{a(p^{k})}{p^{ks}}$

satisfies

$\log F_{p}(s)=\sum_{k=1}^{\infty}\frac{b(p^{k})}{p^{ks}}\quad\text{with
$b(p^{k})\ll p^{\vartheta k}$ for a certain $\vartheta<1/2$.}$ (2.1)

Note that the series in (2.1) is absolutely convergent for $\sigma>\vartheta$
and hence

$F_{p}(s)$ is holomorphic and bounded away from 0 for
$\sigma>\vartheta^{\prime}$ for some $\vartheta^{\prime}<1/2$. (2.2)

We say that $F\in{\mathcal{S}}$ has a polynomial Euler product if for every
prime $p$ there exist $\partial_{p}\in\mathbb{N}$ and
$\alpha_{j,p}\in\mathbb{C}$ such that

$F_{p}(s)=\prod_{j=1}^{\partial_{p}}\left(1-\frac{\alpha_{j,p}}{p^{s}}\right)^{-1}.$

In such a case, it follows from the Ramanujan conjecture that
$|\alpha_{j,p}|\leq 1$; see p.448-449 of [4].

The twist by a Dirichlet character $\chi$ (mod $q$) is defined for $\sigma>1$
as

$F^{\chi}(s)=\sum_{n=1}^{\infty}\frac{a(n)\chi(n)}{n^{s}}.$

We say that $F\in{\mathcal{S}}^{\sharp}$ is weakly twist-regular at $p$ if for
every primitive Dirichlet character $\chi$ (mod $p^{f}$) with $1\leq f\leq
m_{q_{F}}(p)$, where $m_{q_{F}}(p)$ is the order of $p$ (mod $q_{F}$), the
twist $F^{\chi}$ belongs to ${\mathcal{S}}^{\sharp}$ and has the same degree
as $F$. Moreover, the linear twist of $F\in{\mathcal{S}}^{\sharp}$ is defined
for $\sigma>1$ as

$F(s,\alpha)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}e(-n\alpha)$

with $\alpha\in\mathbb{R}$.

Degree $d_{F}$, conductor $q_{F}$, root number $\omega_{F}$ and
$\xi$-invariant $\xi_{F}$ of $F\in{\mathcal{S}}^{\sharp}$ are defined by

$\begin{split}d_{F}=2\sum_{j=1}^{r}\lambda_{j}&,\qquad
q_{F}=(2\pi)^{d}Q^{2}\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},\\\
\omega_{F}=\omega\prod_{j=1}^{r}\lambda_{j}^{-2i\Im(\mu_{j})}&,\qquad\xi_{F}=2\sum_{j=1}^{r}(\mu_{j}-1/2)=\eta_{F}+id_{F}\theta_{F}\end{split}$

with $\eta_{F},\theta_{F}\in\mathbb{R}$. We also write

$\omega_{F}^{*}=\omega_{F}e^{-i\frac{\pi}{2}(\eta_{F}+1)}\big{(}\frac{q_{F}}{(2\pi)^{2}}\big{)}^{i\theta_{F}}\quad\text{and}\quad\tau_{F}=\max_{j=1,\dots,r}\big{|}\frac{\Im(\mu_{j})}{\lambda_{j}}\big{|},$

while $m_{F}$ denotes the order of the pole of $F$ at $s=1$. In
$\omega_{F}^{*}$, and in other definitions below, we changed $\theta_{F}$ to
$2\theta_{F}$ compared to the corresponding definitions in [4]. This is due to
the above slightly different definition of the $\xi$-invariant and the fact
that we are considering functions of degree $d_{F}=2$. The $H$-invariants of
$F$ are defined as

$H_{F}(n)=2\sum_{j=1}^{r}\frac{B_{n}(\mu_{j})}{\lambda_{j}^{n-1}}\hskip
42.67912ptn=0,1,...$

where $B_{n}(x)$ is the $n$-th Bernoulli polynomial. Note that
$H_{F}(0)=d_{F}$ is the degree and $H_{F}(1)$ is the $\xi$-invariant. We refer
to our survey papers [1],[2],[6],[7] for further definitions, examples and the
basic theory of the Selberg class.

As in [4], for $\nu,\mu=1,2,...$ we define the polynomials
$R_{\nu}(s)=R_{\nu,F}(s)$ and $V_{\mu}(s)=V_{\mu,F}(s)$ as

$\begin{split}R_{\nu}(s)&=B_{\nu+1}(1-2s-2i\theta_{F})+B_{\nu+1}(1)\\\
&+\frac{1}{2}\sum_{k=0}^{\nu+1}{\nu+1\choose
k}\big{(}(-1)^{\nu}H_{F}(k)s^{\nu+1-k}-\overline{H_{F}(k)}(1-s)^{\nu+1-k}\big{)}\end{split}$
(2.3)

and

$V_{\mu}(s)=(-1)^{\mu}\sum_{m=1}^{\mu}\frac{1}{m!}\sum_{\begin{subarray}{c}\nu_{1}\geq
1,...,\nu_{m}\geq 1\\\
\nu_{1}+...+\nu_{m}=\mu\end{subarray}}\prod_{j=1}^{m}\frac{R_{\nu_{j}}(s)}{\nu_{j}(\nu_{j}+1)},$
(2.4)

respectively. We also define $Q_{0}(s)\equiv 1$ and, for $\nu=1,2,...$, the
polynomials $Q_{\nu}(s)$ appearing in Theorem 2 by means of the formula

$\exp\big{(}\sum_{\nu=1}^{\infty}\frac{(-1)^{\nu}R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{(w+2s-1+2i\theta_{F})^{\nu}}\big{)}\approx
1+\sum_{\nu=1}^{\infty}\frac{Q_{\nu}(s)}{(w-1)\cdots(w-\nu)},$

and the coefficients $C_{\mu,\ell}$, $\ell\geq\mu\geq 1$, by

$\frac{1}{w^{\mu}}\approx\sum_{\ell=\mu}^{\infty}\frac{C_{\mu,\ell}}{(w-1)\cdots(w-\ell)}.$
(2.5)

Moreover, we define the coefficients $A_{\mu,\nu}(s)$ ($\nu\geq\mu\geq 1$) by

$\frac{1}{(w+2s-1+2i\theta_{F})^{\mu}}\approx\sum_{\nu=\mu}^{\infty}\frac{A_{\mu,\nu}(s)}{(w-1)\cdots(w-\nu)}.$
(2.6)

Here $\approx$ means asymptotic expansion as $w\to\infty$, i.e. cutting the
sum on the left hand side at $\nu=N$ introduces an error of size
$O_{s}(1/|w|^{N+1})$. We refer to Lemmas 8 and 9 below for a more precise
meaning of (2.5) and (2.6).

We write $w=u+iv$ and, for a given $s$, define the contour ${\mathcal{L}}(s)$
as follows:

${\mathcal{L}}(s)={\mathcal{L}}_{-\infty}(s)\cup{\mathcal{L}}_{\infty}(s)$

where

${\mathcal{L}}_{-\infty}(s)=(-\sigma+c_{0}-i\infty,-\sigma+c_{0}+it_{0}]\cup[-\sigma+c_{0}+it_{0},-\sigma-
c_{0}+it_{0}]$ ${\mathcal{L}}_{\infty}(s)=[-\sigma-c_{0}+it_{0},-\sigma-
c_{0}+i\infty).$

Here $t_{0}=t_{0}(s)=c_{1}(|s|+1)^{2}$ and $c_{0},c_{1}>0$ are sufficiently
large constants depending on $F$ to be chosen later on. Observe a significant
difference in the present choice of $t_{0}$ compared to the analogous choice
in [4]; here $t_{0}$ is much larger, and this is important in the proof of
Theorem 2. Moreover, we denote by ${\mathcal{L}}^{*}_{-\infty}(s)$ the half-
line $1-2s-2i\theta_{F}-{\mathcal{L}}_{\infty}(s)$ taken with the positive
orientation, hence

${\mathcal{L}}_{-\infty}^{*}(s)=(1-\sigma+c_{0}-i\infty,1-\sigma+c_{0}-it_{0}^{*}]$

with $t_{0}^{*}=t_{0}^{*}(s)=t_{0}+2t+2\theta_{F}$. Further, we let

${\mathcal{L}}_{\infty}^{*}(s)=[1-\sigma+c_{0}-it_{0}^{*},N+1]\cup[N+1,N+1+i\infty),$

where the positive integer $N$ will be chosen later on (see (5.10) below), and
write

${\mathcal{L}}^{*}(s)={\mathcal{L}}^{*}_{-\infty}(s)\cup{\mathcal{L}}_{\infty}^{*}(s).$

We shall also use the notation

$G(s,w)=\frac{(2\pi)^{1-r}}{\Gamma(1-w)}\prod_{j=1}^{r}\Gamma(\lambda_{j}(1-s-w)+\bar{\mu}_{j})\Gamma(1-\lambda_{j}(s+w)-\mu_{j})$

and

$S(s,w)=\frac{2^{r-1}}{\sin\pi
w}\prod_{j=1}^{r}\sin\big{(}\pi(\lambda_{j}(s+w)+\mu_{j}\big{)}.$

Finally, $A,B,c,c^{\prime},...$ will denote positive constants, possibly
depending on $F$ (also via a dependence on the above constants $c_{0},c_{1}$),
not necessarily the same at each occurrence. The constants in the $\ll$\- and
$O$-symbols may also depend on $F(s)$ (again, also via $c_{0},c_{1}$).

##  3\. Lemmas

######  Lemma 1.

Let $F\in{\mathcal{S}}$ and $q_{F}$ be square-free. Then for $\sigma>1$ we
have

$\sum_{\begin{subarray}{c}1\leq a\leq q_{F}\\\
(a,q_{F})=1\end{subarray}}\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}e(-\frac{a}{q_{F}}n)=F(s)\sum_{d|q_{F}}d\mu(\frac{q_{F}}{d})\prod_{p|d}\left(1-F_{p}(s)^{-1}\right).$
(3.1)

Proof. For simplicity we write $q$ in place of $q_{F}$. Recall the well-known
Kluyver’s formula for the Ramanujan sum

$c_{q}(n):=\sum_{\begin{subarray}{c}1\leq a\leq q\\\
(a,q)=1\end{subarray}}e(an/q)=\sum_{d|(q,n)}\mu(\frac{q}{d})d.$

Since $c_{q}(n)\in\mathbb{R}$, the left hand side of (3.1) equals

$\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}c_{q}(n)=\sum_{d|q}\mu(\frac{q}{d})d^{1-s}\sum_{n=1}^{\infty}\frac{a(dn)}{n^{s}}.$
(3.2)

Now we compute the inner sum for a generic $d|q$, $d=p_{1}\ldots p_{k}$. We
have

$\displaystyle\sum_{n=1}^{\infty}\frac{a(dn)}{n^{s}}$ $\displaystyle=$
$\displaystyle\sum_{\nu_{1}=0}^{\infty}\ldots\sum_{\nu_{k}=0}^{\infty}\sum_{\begin{subarray}{c}n\geq
1\\\ p_{j}^{\nu_{j}}||n\\\ (1\leq j\leq k)\end{subarray}}\frac{a(dn)}{n^{s}}$
$\displaystyle=$
$\displaystyle\sum_{\nu_{1}=0}^{\infty}\ldots\sum_{\nu_{k}=0}^{\infty}\left(\prod_{j=1}^{k}\frac{a(p_{j}^{\nu_{j}+1})}{p_{j}^{\nu_{j}s}}\right)\sum_{(n,p_{1}\ldots
p_{k})=1}\frac{a(n)}{n^{s}}$ $\displaystyle=$
$\displaystyle\sum_{\nu_{1}=0}^{\infty}\ldots\sum_{\nu_{k}=0}^{\infty}\left(\prod_{j=1}^{k}\frac{a(p_{j}^{\nu_{j}+1})}{p_{j}^{\nu_{j}s}}\right)\left(\prod_{j=1}^{k}F_{p_{j}}(s)^{-1}\right)F(s)$
$\displaystyle=$ $\displaystyle
F(s)d^{s}\prod_{j=1}^{k}\left(1-F_{p_{j}}(s)^{-1}\right).$

Inserting this into (3.2) we obtain (3.1), and the proof is complete. ∎

For $F$ in the Selberg class ${\mathcal{S}}$ we denote by $N_{F}(\sigma,T)$
the number of non-trivial zeros $\beta+i\gamma$ of $F$ in the rectangle
$\beta>\sigma$, $|\gamma|\leq T$.

######  Lemma 2.

Let $F\in{\mathcal{S}}$ with $d=2$. Then for every $\varepsilon>0$ and any
fixed $\sigma>1/2$ we have

$N_{F}(\sigma,T)\ll T^{3/2-\sigma+\varepsilon}.$

Proof. See p.474-475 of [4]. ∎

######  Lemma 3.

Let $F\in{\mathcal{S}}^{\sharp}$ with $d=2$. Then there exists a positive
constant $T_{0}=T_{0}(F)$ such that

$F(s)\asymp\left(\frac{q_{F}}{(2\pi
e)^{2}}\right)^{|\sigma|}|s|^{2|\sigma|+1}$

uniformly for $|t|\geq T_{0}$ and $\sigma\leq-1$.

Proof. This is a refined version of Lemma 2.1 in [4]. The proof follows from
the functional equation of $F$ and the Stirling formula; see also Lemma 3 in
[5]. ∎

######  Lemma 4.

Let $F\in{\mathcal{S}}$, $p$ be a prime number and $\sigma>1$. Then there
exist coefficients $c(\chi,p)$, where $\chi$ runs over the Dirichlet
characters $\chi$ (mod $p$), such that

$F(s,1/p)=\sum_{\begin{subarray}{c}\chi(\bmod p)\\\
\chi\neq\chi_{0}\end{subarray}}c(\chi,p)F^{\chi}(s)+\left(1-\frac{p}{p-1}F_{p}(s)^{-1}\right)F(s).$

Proof. See equation (2.6) of [4], observing that $F(s,1/p)=F(s,-(p-1)/p)$. ∎

######  Lemma 5.

Let $F\in{\mathcal{S}}^{\sharp}$, $q$ be a positive integer and let
$\theta<1/2$ be fixed. Moreover, for every prime $p|q$, let $\varepsilon_{p}$
be a complex number with $|\varepsilon_{p}|=1$. Then there exist two positive
constants $a$ and $b$ (depending only on $\theta$) and a sequence of positive
numbers $\tau_{k}\to\infty$, $k\geq 1$, satisfying the following two
conditions

i) for every prime $p|q$ we have $|p^{-i\tau_{k}}-\varepsilon_{p}|<1/k$,

ii) $|F(\sigma+it)|\geq\tau_{k}^{-b}$ uniformly for $-1\leq\sigma\leq\theta$
and $|t-\tau_{k}|\leq\tau_{k}^{a}.$

Proof. The numbers $\log p$, $p|q$ are linearly independent over $\mathbb{Q}$.
Thus by a well known version of the classical Kronecker approximation theorem,
for every $k\geq 1$ there exists a relatively dense set of numbers $\tau$ such
that

$|p^{-i\tau}-\varepsilon_{p}|<\frac{1}{k}\quad\text{for all}\ p|q.$

Thus using Lemma 2 we see that there exists a solution $\tau_{k}\geq
2\max(k,T_{0}(F))$, where $T_{0}(F)$ is the constant appearing in Lemma 3, of
this system of inequalities such that $F(\sigma+it)\neq 0$ for

$-1\leq\sigma<(1+2\theta)/4\ \quad\text{
and}\quad|t-\tau_{k}|\leq\tau_{k}^{a},$

where $a$ is any fixed positive number less than $(1-2\theta)/4$. For $s$ in
such a range we use the well known formula

$-\frac{F^{\prime}}{F}(s)=\sum_{|\gamma-t|<1}\frac{1}{s-\rho}+O(\log\tau_{k}),\quad\text{therefore}\quad-\frac{F^{\prime}}{F}(s)=O(\log\tau_{k}).$

Hence, recalling Lemma 3, we have

$\log|F(\sigma+it)|=\log|F(-1+it)|+\Re\int_{-1}^{\sigma}\frac{F^{\prime}}{F}(u+it)\,du\geq-b\log\tau_{k}$

for $-1\leq\sigma\leq\theta$ and $|t-\tau_{k}|\leq\tau_{k}^{a}$, and the lemma
follows. ∎

######  Lemma 6.

Let $R_{\nu}(s)$ be as in (2.3) and let $c\geq 1$. For $s\in\mathbb{C}$ and
$1\leq\nu\leq c(|s|+1)$ we have

$R_{\nu}(s)\ll\big{(}c^{\prime}(|s|+1)\big{)}^{\nu+1}$

with a suitable $c^{\prime}\geq c+1$.

Proof. This is Lemma 3.8 in [4], with an explicit lower bound for
$c^{\prime}$. ∎

Let

$V_{\mu,N}(s)=(-1)^{\mu}\sum_{m=1}^{\mu}\frac{1}{m!}\sum_{\begin{subarray}{c}1\leq\nu_{1}\leq
N,...,1\leq\nu_{m}\leq N\\\
\nu_{1}+...+\nu_{m}=\mu\end{subarray}}\prod_{j=1}^{m}\frac{R_{\nu_{j}}(s)}{\nu_{j}(\nu_{j}+1)}.$
(3.3)

Recalling definition (2.4) we see that $V_{\mu,N}(s)=V_{\mu}(s)$ for $\mu\leq
N$.

######  Lemma 7.

For $s\in\mathbb{C}$ and $1\leq\mu\leq c(|s|+1)$ we have

$V_{\mu,N}(s)\ll\frac{(c^{\prime}(|s|+1)^{2\mu}}{(\mu-1)!},$

where $c$ and $c^{\prime}$ are as in Lemma 6. Moreover, for all $\mu\geq 1$

$V_{\mu,N}(s)\ll(c^{\prime}(|s|+1)^{2\mu}.$

Proof. Thanks to Lemma 6, for every $\mu\geq 1$ we have

$\begin{split}|V_{\mu,N}(s)|&\leq\sum_{m=1}^{\mu}\frac{1}{m!}\sum_{\nu_{1}+...+\nu_{m}=\mu}\prod_{j=1}^{m}\frac{(c^{\prime}(|s|+1))^{\nu_{j}+1}}{\nu_{j}(\nu_{j}+1)}\\\
&\ll(c^{\prime}(|s|+1))^{\mu}\sum_{m=1}^{\mu}\frac{(c^{\prime}(|s|+1))^{m}}{m!}\ll(c^{\prime}(|s|+1))^{2\mu}.\end{split}$
(3.4)

Moreover, recalling that $c^{\prime}\geq c+1$, for $1\leq\mu\leq c(|s|+1)$ we
have

$\sum_{m=1}^{\mu}\frac{(c^{\prime}(|s|+1))^{m}}{m!}\leq\mu\max_{1\leq
m\leq\mu}\frac{(c^{\prime}(|s|+1))^{m}}{m!}=\frac{(c^{\prime}(|s|+1)^{\mu}}{(\mu-1)!},$

and the lemma follows from this and (3.4). ∎

######  Lemma 8.

For $w\in\mathbb{C}$ and integers $\mu,M$ with $1\leq\mu\leq M\leq|w|/2$ we
have

$\frac{1}{w^{\mu}}=\sum_{\ell=\mu}^{M}\frac{C_{\mu,\ell}}{(w-1)\cdots(w-\ell)}+R_{\mu,M}(w),$
(3.5)

where the coefficients $C_{\mu,\ell}$ are defined by (2.5) and

$R_{\mu,M}(w)\ll\frac{2^{M}M!}{(\mu-1)!}\frac{1}{|w(w-1)\cdots(w-M)|}.$

Proof. This is Lemma 3.13 in [4]. ∎

######  Lemma 9.

Let $s\in\mathbb{C}$ and $1\leq\mu\leq\nu\leq N$ be integers and let

$A_{\mu,\nu}(s)=\sum_{k=0}^{\nu-\mu}{-\mu\choose
k}C_{\mu+k,\nu}(2s-1+2i\theta_{F})^{k}.$

Then for $w\in{\mathcal{L}}_{-\infty}^{*}(s)$ and $1\leq N\leq|\sigma|+c$ we
have

$\frac{1}{(w+2s-1+2i\theta_{F})^{\mu}}=\sum_{\nu=\mu}^{N}\frac{A_{\mu,\nu}(s)}{(w-1)\cdots(w-\nu)}+O\big{(}\frac{4^{N}N!|2s-1+2i\theta_{F}|^{N-\mu+1}}{|w(w-1)\cdots(w-N)|}\big{)},$

where the constant in the $O$-symbol may depend on $c$.

Proof. We follow the proof of Lemma 3.14 in [4], assuming that the constant
$c_{1}$ in the definition of ${\mathcal{L}}_{-\infty}^{*}(s)$ in Section 2 is
sufficiently large and estimating in a different way the error term. Given an
integer $P\geq 0$, for $w\in{\mathcal{L}}_{-\infty}^{*}(s)$ we have that

$\frac{1}{(w+\eta)^{\mu}}=\sum_{k=0}^{P}{-\mu\choose
k}\frac{\eta^{k}}{w^{\mu+k}}+O\left(\frac{2^{P+\mu}|\eta|^{P+1}}{|w|^{P+\mu+1}}\right),$
(3.6)

where $\eta=2s-1+2i\theta_{F}$. Choose $P=N-\mu$. Since $|w|\geq 2N$, for
$w\in{\mathcal{L}}_{-\infty}^{*}$ the error term in (3.6) is

$\ll\frac{4^{N}|\eta|^{N-\mu+1}}{|w(w-1)\cdots(w-N)|}.$ (3.7)

Now we apply Lemma 8 with $M=N$ and replace $1/w^{\mu+k}$ in (3.6) by the main
term of (3.5). This produces an error of size

$\ll\frac{2^{N}N!}{|w(w-1)\cdots(w-N)|}\sum_{k=0}^{N-\mu}\left|{-\mu\choose
k}\right|\frac{|\eta|^{k}}{(k+\mu)!}\ll\frac{4^{N}N!|\eta|^{N-\mu}}{|w(w-1)\cdots(w-N)|},$
(3.8)

hence the expansion of $1/(w+\eta)^{\mu}$ follows from (3.6)–(3.8), and the
lemma is proved. ∎

######  Lemma 10.

Let $s\in\mathbb{C}$ and $c\geq 1$. For $1\leq\nu\leq|s|+c$ we have

$Q_{\nu}(s)\ll\frac{(c^{\prime}(|s|+1))^{2\nu}}{\nu!}$

with a suitable $c^{\prime}=c^{\prime}(c)>0$.

Proof. This is Lemma 3.18 in [4]. ∎

##  4\. Proof of Theorem 1

From Theorem 2 in [5] we know that $F_{p}(s)^{-1}$ is a polynomial in $p^{-s}$
for all primes $p\nmid q_{F}$. The main difficulty lies in proving that the
same holds for the primes $p|q_{F}$.

Let ${\mathbb{T}}=\\{z\in{\mathbb{C}}:|z|=1\\}$ denote the unit circle, $r$ be
the number of prime factors of
$q_{F}$$,{\boldsymbol{\varepsilon}}\in{\mathbb{T}}^{r}$ be given and let
$\tau_{k}$ be the sequence appearing in Lemna 5, with the choice $q=q_{F}$.
Let $l_{k}:=\log\tau_{k}$ and assume without loss of generality that the
$\tau_{k}$’s are sufficiently large. We set

$\Omega_{k}:=\\{s=\sigma+it:\sigma\leq 1/2,|t|\leq l_{k}^{2}\\},$
$\Omega_{k,1}:=\\{s\in\Omega_{k}:\sigma\leq-
l_{k}\\}\quad\text{and}\quad\Omega_{k,2}:=\\{s\in\Omega_{k}:\sigma\geq-
l_{k}\\}.$

Moreover, for $\sigma>1$ let

$G_{k}(s):=\frac{1}{F(s+i\tau_{k})}\sum_{\begin{subarray}{c}1\leq a\leq
q_{F}\\\ (a,q_{F})=1\end{subarray}}F(s+i\tau_{k},a/q_{F}).$ (4.1)

By Lemma 1 we have

$G_{k}(s)=\sum_{d|q_{F}}d\mu(\frac{q_{F}}{d})\prod_{p|d}\left(1-F_{p}(s+i\tau_{k})^{-1}\right)$
(4.2)

hence it follows from (2.2) that

$G_{k}(s)\ll 1\quad\text{for $\sigma>\vartheta^{\prime}$ with a certain
$\vartheta^{\prime}<1/2$}.$ (4.3)

As in the proof of Theorem 3 in [5], for every $a(\bmod q_{F})$,
$(a,q_{F})=1$, we fix a prime $p_{a}\equiv a(\bmod q_{F})$. Obviously
$F(s,a/q_{F})=F(s,p_{a}/q_{F})$, thus we apply Theorem 2 with
$s\in\Omega_{k}$, $\alpha=p_{a}/q_{F}$ and $K=[|\sigma|]+2$ to obtain that

$\displaystyle F$
$\displaystyle\\!\\!\\!\\!\\!(s+i\tau_{k},a/q_{F})=-i\omega_{F}^{*}\left(\frac{p_{a}}{\sqrt{q_{F}}}\right)^{2s-1+2i(\tau_{k}+\theta_{F})}$
(4.4) $\displaystyle\times$
$\displaystyle\sum_{\nu=0}^{K}\left(\frac{ip_{a}}{2\pi}\right)^{\nu}Q_{\nu}(s+i\tau_{k})\overline{F}(s+\nu+i(\tau_{k}+2\theta_{F}),-1/p_{a})+H_{K}(s+i\tau_{k},p_{a}/q_{F}).$

Moreover, using Lemma 4 we have

$\overline{F}(s,-1/p_{a})=\sum_{\begin{subarray}{c}\chi(\bmod p_{a})\\\
\chi\neq\chi_{0}\end{subarray}}\overline{c({\chi},p_{a})}\overline{F^{\chi}}(s)-\left(1-\frac{p_{a}}{p_{a}-1}\overline{F}_{p_{a}}(s)^{-1}\right)\overline{F}(s).$
(4.5)

From Theorem 2 of [5] we know that $\overline{F}_{p_{a}}(s)^{-1}$ is a
polynomial in $p_{a}^{-s}$, and in particular is entire. Since for
$s\in\Omega_{k}$ we have that $\Im(s+\nu+i(\tau_{k}+2\theta_{F}))\neq 0$, and
the possible pole of $F(s)$ and $F^{\chi}(s)$ is at $s=1$, from (4.5) we
deduce that $\overline{F}(s+\nu+i(\tau_{k}+2\theta_{F}),-1/p_{a})$ is
holomorphic for $s\in\Omega_{k}$. Thus from (4.1),(4.4) and ii) of Lemma 5 we
conclude that $G_{k}(s)$ is holomorphic for $s\in\Omega_{k}$.

Now we estimate $G_{k}(s)$ in this region. From (4.1),(4.4) and (4.5) we have

$\begin{split}G_{k}(&s)\ll\frac{B^{|\sigma|}}{|F(s+i\tau_{k})|}\sum_{0\leq\nu\leq
K}|Q_{\nu}(s+i\tau_{k})|\\\ &\times\max_{\begin{subarray}{c}1\leq a\leq
q_{F}\\\ (a,q_{F})=1\end{subarray}}\left(\sum_{\begin{subarray}{c}\chi(\bmod
p_{a})\\\
\chi\neq\chi_{0}\end{subarray}}|\overline{F^{\chi}}(s+\nu+i(\tau_{k}+2\theta_{F}))|+B^{|\sigma+\nu|}|\overline{F}(s+\nu+i(\tau_{k}+2\theta_{F}))|\right)\\\
&+\max_{\begin{subarray}{c}1\leq a\leq q_{F}\\\
(a,q_{F})=1\end{subarray}}\frac{|H_{K}(s+i\tau_{k},p_{a}/q_{F})|}{|F(s+i\tau_{k})|},\end{split}$
(4.6)

where here and later on $B>1$ denotes a certain constant, not necessarily the
same at each occurrence. Note that we used the fact that $F_{p_{a}}(s)^{-1}$
is a polynomial in $p_{a}^{-s}$ and thus it is $\ll B^{|\sigma|}$. Recalling
(1.3), Lemma 3 and our assumption that $F^{\chi}$ is a degree 2 function in
${\mathcal{S}}^{\sharp}$, we see that for $\sigma\leq-1$ the contribution of
the terms with $\nu\leq-\sigma-1$ is at most

$\displaystyle\ll$ $\displaystyle
B^{|\sigma|}\max_{0\leq\nu\leq-\sigma-1}\frac{(|s|+\tau_{k})^{2\nu}}{\nu!}\frac{(|s+\nu+i(\tau_{k}+2\theta_{F})|+1)^{2|\sigma+\nu|+1}}{(|s|+\tau_{k})^{2|\sigma|+1}}$
$\displaystyle\ll$ $\displaystyle
B^{|\sigma|}\max_{0\leq\nu\leq-\sigma-1}\frac{(|\sigma|+\tau_{k})^{2\nu}}{\nu!}\frac{(|\sigma+\nu|+\tau_{k})^{2|\sigma|-2\nu}}{(|\sigma|+\tau_{k})^{2|\sigma|}}$
$\displaystyle\ll$ $\displaystyle
B^{|\sigma|}\max_{0\leq\nu\leq-\sigma-1}\frac{1}{\nu!}\left(\frac{(|\sigma|-\nu+\tau_{k})}{|\sigma|+\tau_{k}}\right)^{2(|\sigma|-\nu)}\ll
B^{|\sigma|}.$

Still for $\sigma\leq-1$, the terms with $-\sigma\leq\nu\leq K$ (there are at
most 3 of them) contribute

$\ll
B^{|\sigma|}\frac{(|\sigma|+\tau_{k})^{2|\sigma|}}{([|\sigma|]+1)!}\frac{\tau_{k}^{A}}{(|\sigma|+\tau_{k})^{2|\sigma|}}\ll
B^{|\sigma|}\frac{\tau_{k}^{A}}{(|\sigma|+1)^{|\sigma|}}\ll\tau_{k}^{A}.$
(4.8)

Finally, for $\sigma\leq-1$, the last term on the right hand side of (4.6)
contributes, recalling also (1.2),

$\frac{H_{K}(s+i\tau_{k},p_{a}/q_{F})}{F(s+i\tau_{k})}\ll\
B^{|\sigma|}\frac{|s+i\tau_{k}|^{2|\sigma|+A}}{(|\sigma|+\tau_{k})^{2|\sigma|+1}}\ll
B^{|\sigma|}(|\sigma|+\tau_{k})^{A}\ll B^{|\sigma|}\tau_{k}^{A}.$ (4.9)

For $s\in\Omega_{k}$ and $\sigma\geq-1$ we have
$|F(s+i\tau_{k})|\gg\tau_{k}^{-A}$ whereas all the other terms in (4.6) are
$\ll\tau_{k}^{A}$. Therefore, gathering (4.6),(4),(4.8) and (4.9) we conclude
that

$G_{k}(s)\ll B^{|\sigma|}\tau_{k}^{A}\quad\text{ for $s\in\Omega_{k}$.}$
(4.10)

Note that the implied constant, as well as $A$ and $B$, may depend on $F$ but
are independent of $\tau_{k}$.

Let now $s\in\Omega_{k,1}$. For such values we have $\tau_{k}^{A}\leq
e^{A|\sigma|}$; hence, recalling our convention on the constant $B$, (4.10)
gives

$G_{k}(s)\ll B^{|\sigma|}\qquad\text{for $s\in\Omega_{k,1}$.}$ (4.11)

For $s\in\Omega_{k,2}$ we write

$g_{k}(s):=\exp(-4\cos(s/(2l_{k})))\quad\text{and}\quad\widetilde{G}_{k}(s):=B^{s}G_{k}(s)g_{k}(s).$

We have $|g_{k}(s)|\leq 1$ for $s\in\Omega_{k,2}$, hence by (4.3) we see that
$\widetilde{G}_{k}(s)\ll 1$ for $s$ on the right vertical part of the boundary
of $\Omega_{k,2}$. Similarly, using (4.11) we see that the same hold for $s$
on the left vertical part of the boundary of $\Omega_{k,2}$. On the horizontal
parts of this boundary we have $g_{k}(s)\ll\exp(-e^{l_{k}/2})$, therefore by
(4.10) we obtain that

$\widetilde{G}_{k}(s)\ll B^{\sigma+|\sigma|}\tau_{k}^{A}\exp(-e^{l_{k}/2})\ll
1.$

Hence, thanks to the maximum modulus principle, we conclude that
$\widetilde{G}_{k}(s)\ll 1$ for $s\in\Omega_{k,2}$. Suppose now that
$s\in\Omega_{k,2}$ and $|t|\leq 1$. Since
$g_{k}(s)\gg\exp(-\exp(O(1/l_{k}))\gg 1$ and $\widetilde{G}_{k}(s)\ll 1$, we
have that $B^{s}G_{k}(s)\ll 1$ for such $s$. This, together with (4.11),
implies that

$B^{s}G_{k}(s)\ll 1\quad\text{for $\sigma\leq 1/2$ and $|t|\leq 1$}$ (4.12)

uniformly on $\tau_{k}$.

Now we recall that the functions $G_{k}(s)$ in (4.1) were defined with the
help of a sequence $\tau_{k}$ in Lemma 5. In particular, we have
$p^{-i\tau_{k}}\to\varepsilon_{p}$ for every $p|q_{F}$. Thus, in view of (2.2)
and (4.2), for $\sigma>\vartheta^{\prime}$ and $|t|\leq 1$ the limit

$G(s,{\boldsymbol{\varepsilon}})=\lim_{k\to\infty}B^{s}G_{k}(s)$ (4.13)

exists and represents a bounded holomorphic function. Hence, by Vitali’s
convergence theorem (see Section 5.21 of Titchmarsh [8]) the limit exists and
$G(s,{\boldsymbol{\varepsilon}})$ is holomorphic for all $\sigma\leq 1/2$,
$|t|\leq 1$. Moreover, recalling (4.12) and i) of Lemma 5 we have that

$G(s,{\boldsymbol{\varepsilon}})\ll 1$ (4.14)

uniformly for ${\boldsymbol{\varepsilon}}\in{\mathbb{T}}^{r}$, $\sigma\leq
1/2$ and $|t|\leq 1$. Writing for $\sigma>\vartheta^{\prime}$

$F_{p}(s)^{-1}=1+\sum_{m=1}^{\infty}\frac{c(p^{m})}{p^{ms}},$

it is easy to see that

$H(s,\varepsilon_{p}):=\lim_{k\to\infty}\left(1-F_{p}(s+i\tau_{k})^{-1}\right)=-\sum_{m=1}^{\infty}\frac{c(p^{m})}{p^{ms}}\varepsilon_{p}^{n}.$

Therefore, recalling (4.2) and (4.13), for $\sigma>\vartheta^{\prime}$ we
obtain

$G(s,{\boldsymbol{\varepsilon}})=B^{s}\sum_{d|q_{F}}d\mu(\frac{q_{F}}{d})\prod_{p|d}H(s,\varepsilon_{p}).$
(4.15)

Let now $d\mu(\boldsymbol{\varepsilon})$ be the Haar measure on
${\mathbb{T}}^{r}=\prod_{p|q_{F}}{\mathbb{T}}_{p}$, where for every $p|q_{F}$
we denote by ${\mathbb{T}}_{p}$ a copy of the unit circle ${\mathbb{T}}$.
Obviously, $d\mu(\boldsymbol{\varepsilon})$ is the product of the normalized
Lebesgue measures $d\mu(\varepsilon_{p})$ on the circles ${\mathbb{T}}_{p}$.
For a given prime $p_{j}|q_{F}$ and a positive integer $m$ let

$J(s)=J(s;p_{j},m):=\int_{{\mathbb{T}}^{r}}G(s,{\boldsymbol{\varepsilon}})\varepsilon_{p_{j}}^{-m}\,d\mu(\boldsymbol{\varepsilon}).$

Since $G(s,{\boldsymbol{\varepsilon}})$ is holomorphic for $\sigma\leq 1/2$
and $|t|\leq 1$, so is $J(s)$. Moreover, recalling (4.14), for such $s$ we
have

$J(s)\ll 1.$ (4.16)

From (4.15), Fubini’s theorem and the orthogonality relations, for
$\sigma>\vartheta^{\prime}$ we have

$\displaystyle J(s)$ $\displaystyle=$ $\displaystyle
B^{s}\sum_{d|q_{F}}d\mu(\frac{q_{F}}{d})\int_{{\mathbb{T}}^{r}}\prod_{p|d}H(s,\varepsilon_{p})\varepsilon_{p_{j}}^{-m}\,d\mu(\boldsymbol{\varepsilon})$
$\displaystyle=$ $\displaystyle
B^{s}\sum_{d|q_{F}}d\mu(\frac{q_{F}}{d})\prod_{\begin{subarray}{c}p|d\\\ p\neq
p_{j}\end{subarray}}\int_{{\mathbb{T}}_{p}}H(s,\varepsilon_{p})\,d\mu(\varepsilon_{p})\int_{{\mathbb{T}}_{p_{j}}}H(s,\varepsilon_{p_{j}})\varepsilon_{p_{j}}^{-m}\,d\mu(\varepsilon_{p_{j}})$
$\displaystyle=$ $\displaystyle
B^{s}p_{j}\mu(\frac{q_{F}}{p_{j}})\int_{{\mathbb{T}}_{p_{j}}}H(s,\varepsilon_{p_{j}})\varepsilon_{p_{j}}^{-m}\,d\mu(\varepsilon_{p_{j}})=B^{s}\mu(\frac{q_{F}}{p_{j}})c(p_{j}^{m})p_{j}^{1-ms}.$

By analytic continuation this equality holds for $\sigma\leq 1/2$ and $|t|\leq
1$, and using (4.16) we obtain

$c(p_{j}^{m})\ll_{p_{j},m}\left(\frac{B}{p_{j}^{m}}\right)^{|\sigma|}.$

For $m$ large enough the right hand side tends to $0$ as $\sigma\to-\infty$,
thus $c(p_{j}^{m})=0$ for such $m$. This shows that $F_{p_{j}}(s)^{-1}$ is a
polynomial in $p_{j}^{-s}$, and Theorem 1 follows. ∎

##  5\. Proof of Theorem 2

We follow closely the proof of Theorem 1.2 in [4], to which we constantly
refer. So here we shall be sketchy, indicating only the main changes. In the
beginning we keep open the value of the sufficiently large constants
$c_{0},c_{1},N$ below, and we add conditions when required. Moreover, we
recall that the path ${\mathcal{L}}(s)$, see Section 2, is defined as in
Section 3.1 of [4] but with a different choice of $t_{0}$; this implies some
differences in the estimates below compared to the analogous estimates in
Section 3.3 of [4].

Let $z_{X}=\frac{1}{X}+2\pi i\alpha$ with a large $X>0$. Writing

$F_{X}(s,\alpha)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}\exp(-nz_{X}),$

as in (3.27) of [4] for $\sigma<2$ we have

$F_{X}(s,\alpha)=\frac{1}{2\pi
i}\int_{(2-\sigma)}F(s+w)\Gamma(w)z_{X}^{-w}\text{d}w=\frac{1}{2\pi
i}\int_{{\mathcal{L}}(s)}F(s+w)\Gamma(w)z_{X}^{-w}\text{d}w.$ (5.1)

If $w=u+iv\in{\mathcal{L}}_{-\infty}(s)$ then $\Re(s+w)\geq-c_{0}$, hence
$F(s+w)\ll|s+w|^{c}$ for some $c>0$ since $F(s)$ has polynomial growth on
vertical strips. If in addition $v<t_{0}$, then $\Re(s+w)=c_{0}>1$ and hence
$F(s+w)\ll 1$. Moreover, still for $w\in{\mathcal{L}}_{-\infty}(s)$, we have

$|z_{X}^{-w}|=|z_{X}|^{-u}\exp(v(\pi/2-\eta(X))),$

where $\eta(X)>0$ and $\eta(X)=O(1/X)$, and by Stirling’s formula

$\Gamma(w)\ll e^{-\frac{\pi}{2}|v|}(|v|+1)^{u-\frac{1}{2}}.$

Therefore, due to the different choice of $t_{0}$, the contribution of
${\mathcal{L}}_{-\infty}(s)$ to (5.1) is

$\frac{1}{2\pi
i}\int_{{\mathcal{L}}_{-\infty}(s)}F(s+w)\Gamma(w)z_{X}^{-w}\text{d}w\ll
A^{|\sigma|}(|s|+1)^{2|\sigma|+c}$

for some positive $A$ and $c$; such constants will not be necessarily the same
at each occurrence. As a consequence, for $\sigma<2$ and any fixed $\alpha>0$,
equation (3.28) of [4] becomes

$\begin{split}F_{X}(s,\alpha)&=\frac{1}{2\pi
i}\int_{{\mathcal{L}}_{\infty}(s)}F(s+w)\Gamma(w)z_{X}^{-w}\text{d}w+O(A^{|\sigma|}(|s|+1)^{2|\sigma|+c})\\\
&={\mathcal{I}}_{X}(s,\alpha)+O(A^{|\sigma|}(|s|+1)^{2|\sigma|+c}),\end{split}$
(5.2)

say, uniformly as $X\to\infty$.

As on p.467 of [4], now we apply the functional equation of $F(s)$ and the
reflection formula of $\Gamma(s)$, thus getting

${\mathcal{I}}_{X}(s,\alpha)=\omega Q^{1-2s}\frac{1}{2\pi
i}\int_{{\mathcal{L}}_{\infty}(s)}\overline{F}(1-s-w)G(s,w)S(s,w)(Q^{2}z_{X})^{-w}\text{d}w;$

see Section 2 for definitions. Taking into account the different choice of
$t_{0}$, following Lemmas 3.1 and 3.2 of [4] for $w\in{\mathcal{L}}_{\infty}$
we have

$S(s,w)=-ie(-\xi_{F}/4)e^{-\pi is}(1+O(e^{-\eta v}))$

for a certain positive $\eta$ and

$G(s,w)\ll e^{-\frac{\pi}{2}(v+2t)}(v+|s|+1)^{2|\sigma|+c}.$

Thus replacing $S(s,w)$ by $-ie(-\xi_{F}/4)e^{-\pi is}$ we obtain

$\begin{split}{\mathcal{I}}_{X}(s,\alpha)&=-i\omega
e(-\xi_{F}/4)Q^{1-2s}e^{-\pi is}\frac{1}{2\pi
i}\int_{{\mathcal{L}}_{\infty}(s)}\overline{F}(1-s-w)G(s,w)(Q^{2}z_{X})^{-w}\text{d}w\\\
&\hskip 28.45274pt+O(A^{|\sigma|}\int_{t_{0}}^{\infty}|G(s,-\sigma-
c_{0}+iv)||(Q^{2}z_{X})^{\sigma+c_{0}-iv}||e^{-\pi is}|e^{-\eta
v}\text{d}v)\\\
&={\mathcal{J}}_{X}(s,\alpha)+O(A^{|\sigma|}(|s|+1)^{2|\sigma|+c}),\end{split}$
(5.3)

say, uniformly as $X\to\infty$.

As on p.467 of [4], we reduce $G(s,w)$ in (5.3) to a single $\Gamma$-factor by
means of the uniform version of the Stirling formula in [3]. For $1\leq
N\leq|s|+c$ and $\beta=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}}$, arguing as
in [4] we obtain

$\begin{split}\log
G(s,w)&=\log\Gamma(1-2s-w-2i\theta_{F})+(\frac{1}{2}-s-w)\log\beta+\log\prod_{j=1}^{r}\lambda_{j}^{-2i\Im\mu_{j}}\\\
&+\sum_{\nu=1}^{N}\frac{R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{w^{\nu}}+O\big{(}\frac{(c(|s|+1))^{N+2}}{|w|^{N+1}}\big{)}.\end{split}$
(5.4)

Since $O\big{(}\frac{(c(|s|+1))^{N+2}}{|w|^{N+1}}\big{)}<1$ for
$w\in{\mathcal{L}}_{\infty}(s)$, provided the constants in the definition of
${\mathcal{L}}_{\infty}(s)$ are sufficiently large, for $1\leq
N\leq|\sigma|+c$ we have

$e^{O\big{(}\frac{(c(|s|+1))^{N+2}}{|w|^{N+1}}\big{)}}=1+O\big{(}A^{|\sigma|}\frac{(|s|+1)^{N+2}}{|w|^{N+1}}\big{)}.$

Hence from (5.4) we obtain

$\begin{split}G(s,w)&=\Gamma(1-2s-w-2i\theta_{F})\beta^{\frac{1}{2}-s-w}\prod_{j=1}^{r}\lambda_{j}^{-2i\Im\mu_{j}}\exp\big{(}\sum_{\nu=1}^{N}\frac{R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{w^{\nu}}\big{)}\\\
&\times\big{(}1+O\big{(}A^{|\sigma|}\frac{(|s|+1)^{N+2}}{|w|^{N+1}}\big{)}\big{)}.\end{split}$
(5.5)

Moreover, using Lemma 6, for $w\in{\mathcal{L}}_{\infty}(s)$ and $1\leq N\leq
c(|s|+1)$ we have

$\begin{split}\sum_{\nu=1}^{N}\frac{R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{w^{\nu}}&\ll\sum_{\nu=1}^{N}\frac{(c^{\prime}(|s|+1))^{\nu+1}}{\nu(\nu+1)}\frac{1}{|w|^{\nu}}\\\
&\ll\sum_{\nu=1}^{N}\frac{1}{\nu(\nu+1)}\frac{(c^{\prime}(|s|+1))^{\nu+1}}{(c_{1}(|s|+1))^{2\nu}}=\sum_{\nu=1}^{N}\frac{1}{\nu(\nu+1)}(\frac{c^{\prime}}{c_{1}})^{2\nu}\ll
1\end{split}$ (5.6)

if $c_{1}\geq c^{\prime}$. Again, we remark that (5.5) and (5.6) hold thanks
to our present choice of $t_{0}$ in the definition of
${\mathcal{L}}_{\infty}(s)$. This small but significant change compared to [4]
leads to much better estimates in the $s$-aspect (compare to Lemma 3.9 and
(3.31) in [4]).

Next we replace $G(s,w)$ by its main term in (5.5) inside the integral
${\mathcal{J}}_{X}(s,\alpha)$ in (5.3). This causes an error of size

$\begin{split}&\ll A^{|\sigma|}e^{\pi
t}\int_{{\mathcal{L}}_{\infty}(s)}|\Gamma(1-2s-w-2i\theta_{F})|\exp\big{(}\Re\big{(}\sum_{\nu=1}^{N}\frac{R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{w^{\nu}}\big{)}\big{)}\big{|}(Q^{2}z_{X})^{-w}\big{|}\frac{(|s|+1)^{N+2}}{|w|^{N+1}}|\text{d}w|\\\
&\ll A^{|\sigma|}e^{\pi
t}(|s|+1)^{N+2}\int_{{\mathcal{L}}_{\infty}(s)}|\Gamma(1-2s-w-2i\theta_{F})|\big{|}(Q^{2}z_{X})^{-w}\big{|}\frac{|dw|}{|w|^{N+1}}\\\
&\ll A^{|\sigma|}(|s|+1)^{N+2}\int_{t_{0}}^{\infty}e^{-\pi v/2}e^{\pi
v/2}v^{-\sigma+c_{0}-N-1/2}\text{d}v\ll A^{|\sigma|}(|s|+1)^{N+2}\end{split}$

if $N\geq-\sigma+c_{0}+1$, the bound being uniform in $X$. Indeed, the bound

$|\Gamma(1-2s-w-2i\theta_{F})(Q^{2}z_{X})^{-w}||w|^{-N-1}\ll
A^{|\sigma|}e^{-\pi t}v^{-\sigma+c_{0}-N-1/2},$

used to obtain the last estimate, follows by an application of Stirling’s
formula, namely

$\begin{split}\Gamma(1-2s-w-2i\theta_{F})&\ll|1-2s-w-2i\theta_{F}|^{1/2-\sigma+c_{0}}\\\
&\hskip 14.22636pt\times\exp((2t+v+2\theta_{F})\arg(1-2s-w-2i\theta_{F}))\\\
&\ll
v^{1/2-\sigma+c_{0}}\exp((2t+v+2\theta_{F})\arg(1-2s-w-2i\theta_{F})),\end{split}$

observing that

$\arg(1-2s-w-2i\theta_{F})=-\arctan\left(\frac{2t+v+2\theta_{F}}{1-2\sigma-u}\right)=-\frac{\pi}{2}+O\left(\frac{|\sigma|+1}{2t+v+2\theta_{F}}\right).$

Therefore, from (5.3), (5.5) and recalling the definition of $q_{F}$ and
$\omega^{*}_{F}$, for

$|\sigma|+c\leq N\leq|\sigma|+c+1$ (5.7)

we have

$\begin{split}{\mathcal{J}}_{X}(s,\alpha)&=\omega^{*}_{F}\big{(}\frac{q_{F}}{4\pi^{2}}\big{)}^{1/2-s-i\theta_{F}}e(-\frac{1}{2}(s+i\theta_{F}))\frac{1}{2\pi
i}\int_{{\mathcal{L}}_{\infty}(s)}\overline{F}(1-s-w)\ \times\\\
&\qquad\times\Gamma(1-2s-w-2i\theta_{F})\exp\big{(}\sum_{\nu=1}^{N}\frac{R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{w^{\nu}}\big{)}\big{(}\frac{q_{F}z_{X}}{4\pi^{2}}\big{)}^{-w}\text{d}w+O\big{(}A^{|\sigma|}(|s|+1)^{2|\sigma|+c^{\prime}}\big{)}.\end{split}$

Hence, by the substitution $1-2s-w-2i\theta_{F}\to w$ in the above integral
and recalling the definition of ${\mathcal{L}}^{*}_{-\infty}(s)$ in Section 2
we obtain

$\begin{split}{\mathcal{J}}_{X}&(s,\alpha)=-i\omega^{*}_{F}\big{(}\sqrt{q_{F}}\alpha-i\frac{\sqrt{q_{F}}}{2\pi
X}\big{)}^{2s-1+2i\theta_{F}}\frac{1}{2\pi
i}\int_{{\mathcal{L}}^{*}_{-\infty}(s)}\bar{F}(s+w+2i\theta_{F})\Gamma(w)\\\
&\times\exp\big{(}\sum_{\nu=1}^{N}\frac{(-1)^{\nu}R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{(w+2s-1+2i\theta_{F})^{\nu}}\big{)}\big{(}\frac{q_{F}z_{X}}{4\pi^{2}}\big{)}^{w}\text{d}w+O\big{(}A^{|\sigma|}(|s|+1)^{2|\sigma|+c^{\prime}})\big{)},\end{split}$
(5.8)

uniformly as $X\to\infty$.

Writing again $\eta=2s-1+2i\theta_{F}$, from the power series expansion of the
exponential function and recalling (3.3) we have

$\begin{split}\exp\big{(}\sum_{\nu=1}^{N}\frac{(-1)^{\nu}R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{(w+\eta)^{\nu}}\big{)}&=1+\sum_{\mu=1}^{\infty}\frac{V_{\mu,N}(s)}{(w+\eta)^{\mu}}\\\
&=1+\sum_{\mu=1}^{N}\frac{V_{\mu}(s)}{(w+\eta)^{\mu}}+O(\sum_{\mu=N+1}^{\infty}\frac{|V_{\mu,N}(s)|}{|w+\eta|^{\mu}}).\end{split}$
(5.9)

Since for $w\in{\mathcal{L}}^{*}_{-\infty}(s)$ we have $|w+\eta|\geq
2(c^{\prime}(|s|+1))^{2}$, where $c^{\prime}$ denotes the constant in Lemma 7,
using such a lemma we obtain

$\begin{split}\sum_{\mu=N+1}^{\infty}\frac{|V_{\mu,N}(s)|}{|w+\eta|^{\mu}})&\ll\sum_{\mu=N+1}^{\infty}\frac{(c^{\prime}(|s|+1)^{2\mu}}{|w+\eta|^{\mu}}\leq\frac{(c^{\prime}(|s|+1)^{2(N+1)}}{|w+\eta|^{N+1}}\sum_{\mu=N+1}^{\infty}2^{-\mu}\\\
&\ll A^{|\sigma|}\frac{(|s|+1)^{2N+1}}{|w(w-1)\cdots(w-N)|}.\end{split}$

Thus using (5.9) and Lemmas 7 and 9

$\begin{split}&\exp\big{(}\sum_{\nu=1}^{N}\frac{(-1)^{\nu}R_{\nu}(s)}{\nu(\nu+1)}\frac{1}{(w+\eta)^{\nu}}\big{)}=1+\sum_{\mu=1}^{N}\frac{V_{\mu}(s)}{(w+\eta)^{\mu}}+O(A^{|\sigma|}\frac{(|s|+1)^{2N+1}}{|w(w-1)\cdots(w-N)|})\\\
&=1+\sum_{\mu=1}^{N}V_{\mu}(s)\sum_{\nu=\mu}^{N}\frac{A_{\mu,\nu}(s)}{(w-1)\cdots(w-\nu)}+O\big{(}\frac{4^{N}N!}{|w(w-1)\cdots(w-N)|}\sum_{\mu=1}^{N}|V_{\mu}(s)||\eta|^{N-\mu+1}\big{)}\\\
&\hskip
182.1207pt+O(A^{|\sigma|}\frac{(|s|+1)^{2N+1}}{|w(w-1)\cdots(w-N)|})\\\
&=\sum_{\nu=0}^{N}\frac{Q_{\nu}(s)}{(w-1)\cdots(w-\nu)}+O\big{(}\frac{4^{N}N!(c^{\prime}(|s|+1)^{N}}{|w(w-1)\cdots(w-N)|}\sum_{\mu=1}^{N}\frac{(c^{\prime}(|s|+1)^{\mu+1}}{(\mu-1)!}\big{)}\\\
&\hskip 130.08621pt\
+O(A^{|\sigma|}\frac{(|s|+1)^{2N+1}}{|w(w-1)\cdots(w-N)|})\\\
&=\sum_{\nu=0}^{N}\frac{Q_{\nu}(s)}{(w-1)\cdots(w-\nu)}+O(A^{|\sigma|}\frac{(|s|+1)^{2N+1}}{|w(w-1)\cdots(w-N)|})\end{split}$

Now we replace the term $\exp(\sum_{\nu=1}^{N}...)$ in (5.8) by the main term
of the above formula. This causes a further error of size

$\begin{split}&\ll
A^{|\sigma|}(|s|+1)^{2N+1}\int_{{\mathcal{L}}^{*}_{-\infty}(s)}\frac{|\Gamma(w)|}{|w(w-1)\cdots(w-N)|}e^{\pi|v|/2}|\text{d}w|\\\
&=A^{|\sigma|}(|s|+1)^{2N+1}\int_{{\mathcal{L}}^{*}_{-\infty}(s)}\frac{|\Gamma(w-N)|e^{\pi|v|/2}}{|w|}|\text{d}w|.\end{split}$

Moreover, we also want $N$ such that $\Re(w-N)\leq 0$ for
$w\in{\mathcal{L}}^{*}_{-\infty}(s)$, i.e. we choose

$N=[-\sigma]+k$ (5.10)

with a sufficiently large positive integer $k$ satisfying (5.7). With such a
choice of $N$ we have $|\Gamma(w-N)|\ll e^{-\pi|v|/2}/|v|^{1/2}$, hence the
integral is $\ll 1$. Therefore (5.8) becomes, uniformly as $X\to\infty$,

$\begin{split}{\mathcal{J}}_{X}(s,\alpha)&=-i\omega^{*}_{F}\big{(}\sqrt{q_{F}}\alpha-i\frac{\sqrt{q_{F}}}{2\pi
X}\big{)}^{2s-1+2i\theta_{F}}\sum_{\nu=0}^{N}Q_{\nu}(s)\frac{1}{2\pi
i}\int_{{\mathcal{L}}^{*}_{-\infty}(s)}\overline{F}(s+w+2i\theta_{F})\
\times\\\
&\qquad\times\Gamma(w-\nu)\big{(}\frac{q_{F}z_{X}}{4\pi^{2}}\big{)}^{w}\text{d}w+O\big{(}A^{|\sigma|}(|s|+1)^{2|\sigma|+B}),\end{split}$
(5.11)

which is the analog of (3.35) in [4].

Replacing the path of integration in (5.11) by the whole path
${\mathcal{L}}^{*}(s)$ causes an error which, since $\nu\leq N$, by Lemma 10
is of size

$\ll
A^{|\sigma|}\sum_{\nu=0}^{N}\frac{(|s|+1)^{2\nu}}{\nu!}\int_{{\mathcal{L}}_{\infty}^{*}(s)}|F(s+w+2i\theta_{F})\Gamma(w-\nu)\big{(}\frac{q_{F}z_{X}}{4\pi^{2}}\big{)}^{w}||\text{d}w|.$

Let now $w\in{\mathcal{L}}^{*}_{\infty}(s)$. Clearly, $F(s+w+2i\theta_{F})\ll
1$. Moreover, for $0\leq\nu\leq N$ we have $-c\leq\Re(w-\nu)\leq N+1-\nu$ and
hence for $v\neq 0$

$v\arg(w-\nu)=v\arg(iv)+\arg\left(1+i\frac{\nu-u}{v}\right)=\frac{\pi}{2}|v|+O(|\sigma|+1).$

Observing that this estimate holds for $v=0$ as well, by Stirling’s formula we
get

$\log|\Gamma(w-\nu)|=(u-\nu-1/2)\log|w-\nu|-\frac{\pi}{2}|v|+O(|\sigma|+1).$

Consequently

$\Gamma(w-\nu)\ll A^{|\sigma|}e^{-\frac{\pi}{2}|v|}(|s|+|v|+1)^{N-\nu+1/2}.$

Further, $|z_{X}^{w}|\ll A^{|\sigma|}e^{-v\arg z_{X}}$, thus the above
mentioned error is (with a suitable $c^{\prime\prime}>0$)

$\ll
A^{|\sigma|}\sum_{\nu=0}^{N}\frac{(|s|+1)^{2\nu}}{\nu!}\int_{-t_{0}^{*}(s)}^{\infty}(|s|+|v|+1)^{|\sigma|-\nu+c^{\prime\prime}}e^{-\pi|v|/2-v\arg{z_{X}}}\text{d}v\ll
A^{|\sigma|}(|s|+1)^{2|\sigma|+B}.$

We also have by Cauchy’s theorem that for $0\leq\nu\leq N$

$\begin{split}&\frac{1}{2\pi
i}\int_{{\mathcal{L}}^{*}(s)}\overline{F}(s+w+2i\theta_{F})\Gamma(w-\nu)\big{(}\frac{q_{F}z_{X}}{4\pi^{2}}\big{)}^{w}\text{d}w\\\
&\hskip-14.22636pt=\frac{1}{2\pi
i}\int_{|\sigma|+\nu+2-i\infty}^{|\sigma|+\nu+2+i\infty}\overline{F}(s+w+2i\theta_{F})\Gamma(w-\nu)\big{(}\frac{q_{F}z_{X}}{4\pi^{2}}\big{)}^{w}\text{d}w\\\
&\hskip-14.22636pt=\big{(}\frac{q_{F}}{4\pi^{2}X}+i\frac{q_{F}\alpha}{2\pi}\big{)}^{\nu}\sum_{n=1}^{\infty}\frac{\overline{a(n)}}{n^{s+\nu+2i\theta_{F}}}\exp(-\frac{4\pi^{2}}{q_{F}z_{X}}n)\end{split}$

since the poles of the integrand lie to the left of ${\mathcal{L}}^{*}(s)$.
Consequently, (5.11) becomes

$\begin{split}{\mathcal{J}}_{X}(s,\alpha)&=-i\omega^{*}_{F}\big{(}\sqrt{q_{F}}\alpha-i\frac{\sqrt{q_{F}}}{2\pi
X}\big{)}^{2s-1+2i\theta_{F}}\sum_{\nu=0}^{N}\big{(}\frac{q_{F}}{4\pi^{2}X}+i\frac{q_{F}\alpha}{2\pi}\big{)}^{\nu}Q_{\nu}(s)\\\
&\hskip
22.76228pt\times\sum_{n=1}^{\infty}\frac{\overline{a(n)}}{n^{s+\nu+2i\theta_{F}}}\exp(-\frac{4\pi^{2}}{q_{F}z_{X}}n)+O(A^{|\sigma|}(|s|+1)^{2|\sigma|+B}),\end{split}$
(5.12)

uniformly as $X\to\infty$. Note that, since

$-\frac{4\pi^{2}}{q_{F}z_{X}}=\frac{2\pi
i}{q_{F}\alpha}-\frac{1}{q_{F}\alpha^{2}}\frac{1}{X+O(1)},$

the series in (5.12) is absolutely convergent for all $s$, for every $\nu$.

As on p.472 of [4], the final step is to make the range of summation of $\nu$
in (5.12) independent of $\sigma$ (recall that $N$ depends on $\sigma$, see
(5.10)). Let $K>0$ be a large integer and $\sigma>-K+1/2$. Depending on the
relative sizes of $N$ and $K$, we add to or withdraw from (5.12) the terms
with $\nu$ between $N+1$ and $K$ or between $K+1$ and $N$, respectively. In
both cases we have that $\sigma+\nu>3/2$ for such $\nu$’s (call them
$\nu\in{\mathcal{X}}$), hence from Lemma 10 we deduce that

$\begin{split}-i\omega^{*}_{F}&\big{(}\sqrt{q_{F}}\alpha-i\frac{\sqrt{q_{F}}}{2\pi
X}\big{)}^{2s-1+2i\theta_{F}}\sum_{\nu\in{\mathcal{X}}}\big{(}\frac{q_{F}}{4\pi^{2}X}+i\frac{q_{F}\alpha}{2\pi}\big{)}^{\nu}Q_{\nu}(s)\times\\\
&\times\sum_{n=1}^{\infty}\frac{\overline{a(n)}}{n^{s+\nu+2i\theta_{F}}}\exp(-\frac{4\pi^{2}}{q_{F}z_{X}}n)\ll
A^{|\sigma|}\sum_{\nu\in{\mathcal{X}}}\frac{(c^{\prime}(|s|+1))^{2\nu}}{\nu!}\end{split}$
(5.13)

uniformly in $X$. If $N<K$ this is

$\ll A^{|\sigma|}(|s|+1)^{2K}\sum_{\nu\geq N+1}\frac{c^{\prime
2\nu}}{\nu!}\ll\frac{A^{|\sigma|}c^{\prime
2N}}{(N+1)!}(|s|+1)^{2K}\sum_{\nu\geq 0}\frac{c^{\prime
2\nu}}{\nu!}\ll(|s|+1)^{2K},$

while if $N>K$ we have

$A^{|\sigma|}\sum_{\nu\in{\mathcal{X}}}\frac{(c^{\prime}(|s|+1))^{2\nu}}{\nu!}\ll
A^{|\sigma|}\frac{(c^{\prime}(|s|+1)^{2N}}{(N-1)!}\ll(|s|+1)^{2N}\ll(|s|+1)^{2K+A}.$

Thus, in view of (5.2),(5.3),(5.12) and (5.13), for $-K+1/2<\sigma<2$ we have

$\begin{split}F_{X}(s,\alpha)=&-i\omega^{*}_{F}\big{(}\sqrt{q_{F}}\alpha-i\frac{\sqrt{q_{F}}}{2\pi
X}\big{)}^{2s-1+2i\theta_{F}}\sum_{\nu=0}^{K}\big{(}\frac{q_{F}}{4\pi^{2}X}+i\frac{q_{F}\alpha}{2\pi}\big{)}^{\nu}\\\
&\times
Q_{\nu}(s)F^{*}_{X}(s+\nu+2i\theta_{F},\alpha)+H_{X}(s,\alpha),\end{split}$
(5.14)

where

$F^{*}_{X}(s,\alpha)=\sum_{n=1}^{\infty}\frac{\overline{a(n)}}{n^{s}}\exp(-\frac{4\pi^{2}}{q_{F}z_{X}}n)$

and

$H_{X}(s,\alpha)\ll(|s|+1)^{2K+A}$

uniformly as $X\to\infty$. Moreover, since $F_{X}(s,\alpha)$,
$F^{*}_{X}(s,\alpha)$ and $Q_{\nu}(s)$ are entire functions, $H_{X}(s,\alpha)$
is also entire. Further, from (5.14) we have that for $1<\sigma<2$

$\lim_{X\to\infty}H_{X}(s,\alpha)=H(s,\alpha)$

exists and is holomorphic since this is clearly true for $F_{X}(s,\alpha)$ and
$F^{*}_{X}(s,\alpha)$. Thanks to (5.14), for $1<\sigma<2$ we also have that

$H(s,\alpha)=F(s,\alpha)+i\omega^{*}_{F}\big{(}\sqrt{q_{F}}\alpha\big{)}^{2s-1+2i\theta_{F}}\sum_{\nu=0}^{K}\big{(}i\frac{q_{F}\alpha}{2\pi}\big{)}^{\nu}Q_{\nu}(s)\bar{F}(s+\nu+2i\theta_{F},-\frac{1}{q_{F}\alpha}).$
(5.15)

Hence by Vitali’s convergence theorem the limit function $H(s,\alpha)$ exists
and is holomorphic for $-K+1/2<\sigma<2$, and satisfies

$H(s,\alpha)\ll(|s|+1)^{2K+A}.$

This provides analytic continuation and bounds for the right-hand side of
(5.15), and Theorem 2 follows.

## References

  * [1] J.Kaczorowski - Axiomatic theory of $L$-functions: the Selberg class \- In Analytic Number Theory, C.I.M.E. Summer School, Cetraro (Italy) 2002, ed. by A.Perelli and C.Viola, 133–209, Springer L.N. 1891, 2006.
  * [2] J.Kaczorowski, A.Perelli - The Selberg class: a survey \- In Number Theory in Progress, Proc. Conf. in Honor of A.Schinzel, ed. by K.Györy et al., 953–992, de Gruyter 1999.
  * [3] J.Kaczorowski, A.Perelli - A uniform version of Stirling’s formula \- Funct. et Approx. 45 (2011), 89–96.
  * [4] J.Kaczorowski, A.Perelli - Twists, Euler products and a converse theorem for $L$-functions of degree $2$ \- Annali Scuola Normale Sup. Pisa (V) 14 (2015), 441–480.
  * [5] J. Kaczorowski, A. Perelli - Twists by Dirichlet characters and polynomial Euler products of $L$-functions \- Preprint.
  * [6] A.Perelli - A survey of the Selberg class of $L$-functions, part I \- Milan J. Math. 73 (2005), 19–52.
  * [7] A.Perelli - A survey of the Selberg class of $L$-functions, part II \- Riv. Mat. Univ. Parma (7) 3* (2004), 83–118.
  * [8] E.C.Titchmarsh - Theory of Functions \- Oxford U.P. 1952.

Jerzy Kaczorowski, Faculty of Mathematics and Computer Science, A.Mickiewicz
University, 61-614 Poznań, Poland and Institute of Mathematics of the Polish
Academy of Sciences, 00-956 Warsaw, Poland. e-mail<EMAIL_ADDRESS>
Alberto Perelli, Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, Italy. e-mail<EMAIL_ADDRESS>


# Search for Black Hole Merger Families

Doğa Veske Department of Physics, Columbia University in the City of New York,
New York, NY 10027, USA  Andrew G. Sullivan Department of Physics, Columbia
University in the City of New York, New York, NY 10027, USA  Zsuzsa Márka
Columbia Astrophysics Laboratory, Columbia University in the City of New York,
New York, NY 10027, USA  Imre Bartos Department of Physics, University of
Florida, PO Box 118440, Gainesville, FL 32611-8440, USA  K. Rainer Corley
Department of Physics, Columbia University in the City of New York, New York,
NY 10027, USA  Columbia Astrophysics Laboratory, Columbia University in the
City of New York, New York, NY 10027, USA  Johan Samsing Niels Bohr
International Academy, The Niels Bohr Institute, Blegdamsvej 17, DK-2100,
Copenhagen, Denmark  Riccardo Buscicchio Institute for Gravitational Wave
Astronomy & School of Physics and Astronomy, University of Birmingham,
Birmingham, B15 2TT, UK  Szabolcs Márka Department of Physics, Columbia
University in the City of New York, New York, NY 10027, USA

###### Abstract

The origin, environment, and evolution of stellar-mass black hole binaries are
still a mystery. One of the proposed binary formation mechanisms is manifest
in dynamical interactions between multiple black holes. A resulting framework
of these dynamical interactions is the so-called hierarchical triple merger
scenario, which happens when three black holes become gravitationally bound,
causing two successive black hole mergers to occur. In such successive
mergers, the black holes involved are directly related to each other, and
hence this channel can be directly tested from the properties of the detected
binary black hole mergers. Here we present a search for hierarchical triple
mergers among events within the GWTC-1 and GWTC-2 catalogs of LIGO/Virgo, the
eccentric localization of GW190521 and those found by the IAS-Princeton group.
The search includes improved statistical quantification that also accounts for
black hole spins. We perform our analysis for different upper bounds on the
mass distribution of first generation BHs. Our results demonstrate the
importance of the mass distributions’ properties for constraining the
hierarchical merger scenario. We present the individually significant merger
pairs. The search yields interesting candidate families and hints of its
future impact.

Astrophysical black holes (98) – Gravitational wave astronomy (675) –
Gravitational wave sources (677)

##  1 Introduction

With the gravitational-wave (GW) detectors LIGO (Aasi et al., 2015) and Virgo
(Acernese et al., 2014) reaching necessary sensitivities for recognizing
binary black hole (BBH) merger signals (Abbott et al., 2016), an increasing
number of detected BBH mergers are being collected. The GW strain data has
been analyzed by LIGO Scientific and Virgo Collaborations and GW catalogues
have been released (Abbott et al., 2019a, 2020b). Since the data was made
public (Abbott et al., 2021), individual groups have found additional BBH
mergers as well (Venumadhav et al., 2020; Zackay et al., 2019a; Nitz et al.,
2019; Zackay et al., 2019b). As the number of detections increases, a wider
array of unique mergers are detected which provide tests of general relativity
and its alternatives at different conditions (Abbott et al., 2019b, 2020c)

Another piece of information that can be acquired from gravitational wave
bursts relates to the interactions of black holes (BH) with each other as
there are various proposed formation channels for inspiraling BBH systems
(Rodriguez et al., 2016c; Zevin et al., 2017; Yang et al., 2019; Samsing et
al., 2014; Samsing & Ramirez-Ruiz, 2017; Samsing et al., 2018b; Samsing, 2018;
Samsing et al., 2018a; Samsing & D’Orazio, 2018a; Zevin et al., 2019;
Rodriguez et al., 2018; Samsing et al., 2020, 2019; Vigna-Gómez et al., 2020;
Fragione & Kocsis, 2019). Available formation scenarios include field binaries
(Dominik et al., 2012, 2013, 2015; Belczynski et al., 2016b, a; Silsbee &
Tremaine, 2017; Murguia-Berthier et al., 2017; Rodriguez & Antonini, 2018;
Schrøder et al., 2018), chemically homogeneous binary evolution (De Mink &
Mandel, 2016; Mandel & de Mink, 2016; Marchant, Pablo et al., 2016), dynamical
mergers in dense stellar clusters (Portegies Zwart & McMillan, 2000; Banerjee
et al., 2010; Tanikawa, 2013; Bae et al., 2014; Rodriguez et al., 2015, 2016a,
2016b; Askar et al., 2017; Park et al., 2017; Samsing, 2018; Samsing &
D’Orazio, 2018b), in active galactic nuclei (AGN) discs (Bartos et al., 2017b,
a; McKernan et al., 2018; Gayathri et al., 2020a; Graham et al., 2020; Yang et
al., 2019; Corley et al., 2019; McKernan et al., 2019; Stone et al., 2017;
McKernan et al., 2012, 2014; Bellovary et al., 2016; Samsing et al., 2020;
Yang et al., 2019; Abbott et al., 2020d; Yang et al., 2020) and in galactic
nuclei (GN) (O’Leary et al., 2009; Hong & Lee, 2015; VanLandingham et al.,
2016; Antonini & Rasio, 2016; Stephan et al., 2016; Hoang et al., 2018; Hamers
et al., 2018; Fragione et al., 2019), very massive stellar mergers (Loeb,
2016; Woosley, 2016; Janiuk et al., 2017; D’Orazio & Loeb, 2018) and single-
single GW captures of primordial black holes (Bird et al., 2016; Cholis et
al., 2016; Sasaki et al., 2016; Carr et al., 2016).

One specific BBH merger scenario of interest is the so-called hierarchical
triple merger, which is proposed to occur in the dynamical BBH formation
channels (Samsing & Ilan, 2018, 2019). Hierarchical triple mergers can occur
through the interaction of three BHs $(BH_{1},BH_{2},BH_{3})$ which form a
gravitationally bound three-body system. This three-body interaction
facilitates the inspiral of the BHs in the system (Campanelli et al., 2008;
Lousto & Zlochower, 2008). First two BHs merge emitting a GW signal, leaving
behind a BBH composed of the first merger remnant $(BH_{1},BH_{2}\rightarrow
BH_{12})$. It eventually merges with the remaining single BH $(BH_{3})$ to
produce a second GW signal $(BH_{12},BH_{3}\rightarrow BH_{123})$ (Samsing &
Ilan, 2019). The process is depicted in Fig. 1

Figure 1: Depiction of a triple hierarchical merger, for a bound 3 BH system.
We show the relationship between the generation of BHs with their colors.
Arrows indicate the chronological order.

Under certain orbital configurations, it has been shown that both mergers can
be observed within timescales of about few years (Samsing & Ilan, 2019). In
previous work (Veske et al., 2020), this scenario has been observationally
constrained by using the mergers in the first GW transient catalog (GWTC-1)
(Abbott et al., 2019a) and the mergers found by the IAS-Princeton group
(Venumadhav et al., 2020; Zackay et al., 2019a, b) from LIGO/Virgo’s first and
second observing runs (O1 and O2).

In this paper, we extend our previous search (Veske et al., 2020) by including
the BBH mergers (except the single detector detection GW190424) from the
second GW transient catalog (GWTC-2) (Abbott et al., 2020b) published by LIGO
Scientific Collaboration and Virgo Collaboration which includes the mergers
from the first half of LIGO and Virgo detectors’ third observing run (O3a) and
the eccentric localization of GW190521 (Gayathri et al., 2020b). In addition,
we improve our test statistic by using the BH spins estimated for each merger.

This Letter is organized as follows; in Sec. 2 we describe the details of our
search. In Sec. 3 we present our results and provide discussion. In Sec. 4 we
conclude.

##  2 Search

Our search analyzes pairs of BBH mergers, using a number of parameters which
behave differently under the two hypotheses under consideration: the two
mergers are unrelated and the two mergers are related through the hierarchical
triple merger scenario. For hierarchical triple mergers, the mass and spin of
the remnant BH from the first merger must correspond to the mass and spin of
one of the BHs in the second merger. In addition, the sky localizations of the
first and second mergers must be overlapping. In the case of unrelated
mergers, the location of two mergers and the masses and spins of their
component BHs are independent of each other. Our search is based on a
frequentist $p$-value assignment through the use of a test statistic (TS). As
Neyman-Pearson’s lemma suggests (Neyman & Pearson, 1933), we choose our TS to
be the likelihood ratio of the _signal_ hypothesis $H_{s}$ – a hierarchical
triple merger, and the _null_ hypothesis $H_{0}$ \- two unrelated mergers.

###  2.1 Merger Properties

Let merger #1 be the merger whose remnant then participates in the second
merger, which will be denoted as merger #2. We use four properties of each BBH
merger for calculating the likelihood ratios, which we list below.

  * •

Correct time order: For hierarchical triples, merger #1 must occur before
merger #2.

  * •

Mass estimates: The mass of the remnant of merger #1 should be consistent with
the mass of one of the BHs in merger #2. For unrelated mergers the masses are
assumed to be independently drawn from the initial BH mass distribution. Both
theoretical and empirical estimates of the initial BH mass distributions are
uncertain. Therefore we consider several different mass distributions as
explained in Sec. 2.2.

  * •

Effective spin parameter $\chi_{\text{eff}}$: For assessing consistency
between the remnant of merger #1 and the BHs in merger #2, beyond mass we also
consider the BHs spin. Remnants from BBH mergers are expected to be highly
spinning, typically around the dimensionless spin magnitude
$|\vec{a}|=|c\vec{S}/Gm^{2}|\sim 0.7$ (Berti & Volonteri, 2008), where $c$ is
the speed of light, $\vec{S}$ is the spin angular momentum of the BH, $G$ is
the universal gravitational constant and $m$ is the mass of the BH. For spin
comparison we use the effective spin parameter $\chi_{\text{eff}}$ which is
the best constrained by gravitational waves measurements. We use
$\chi_{\text{eff}}$ instead of the actual spins due to the fact that parameter
estimations of $\chi_{\text{eff}}$ can be approximated as normal
distributions, where as spin distributions cannot be approximated with a
simple parameterization. We elaborate on it more in Sec. 2.3.
$\chi_{\text{eff}}$ is defined as

$\chi_{\text{eff}}=\frac{\vec{a_{1}}m_{1}+\vec{a_{2}}m_{2}}{m_{1}+m_{2}}\cdot\hat{L}$
(1)

where $m_{1,2}$ are the masses of the initial BHs in the merger,
$\vec{a_{1,2}}$ are their dimensionless spin vectors and $\hat{L}$ is the unit
vector of the binary’s orbital angular momentum. We assume an initial BH
dimensionless spin magnitude to be uniform between [0,1]. We further assume
that in BBH mergers spin orientations are random following a uniform isotropic
distribution, which is expected for dynamical interactions (Vitale et al.,
2017), but not from isolated binaries (e.g. Bogdanović et al. (2007)).

  * •

Localization: The localizations of mergers #1 and #2 must be consistent with
each other. We assume them to happen at the same point in space and neglect
possible travel between the mergers. We assume that the merger rate is uniform
in comoving volume.

  * •

Eccentricity: In triple hierarchical merger scenario, especially for merger
#1, eccentric mergers are expected (Samsing & Ilan, 2019). However, we do not
use the eccentricity in our test statistic due to the following reasoning. The
most sensitive BBH merger searches are template based matched filter searches
which use circular orbits (Abbott et al., 2020b), except the very recent work
targeted on GW190521 (Gayathri et al., 2020b). Unmodeled burst searches can
catch the eccentric mergers but with a lower reach; however, no such merger
has been detected by unmodeled searches with high significance (Abbott et al.,
2019c). Due to the lack of sufficient eccentric waveforms and their
computationally expensive analysis, the precise estimation of the eccentricity
of the detected mergers cannot be done too. Nevertheless, there are
eccentricity estimations for the detected mergers assuming low eccentricity,
and events in GWTC-1 were found to have low eccentricity ($<0.1$) (Romero-Shaw
et al., 2019). On the other hand, the event GW190521 is claimed to have high
eccentricity by individual parties (Gayathri et al., 2020b; Romero-Shaw et
al., 2020). We add the localization of the best matching eccentric template of
GW190521 (Gayathri et al., 2020b) to our analysis, as the only available
eccentric localization.

###  2.2 BH mass distributions

An important part of the analysis is the mass distribution for first
generation BHs. Although there is no constraint on the mass of BHs from
general relativity, considering the formations channels from stellar
evolution, a restricted region (known as the upper mass gap) appears between
$\sim 50-135M_{\odot}$ for first generation BHs due to phenomena known as pair
instability supernova (PISN) and pulsational pair instability supernova
(PPISN) (Woosley, 2017). Both are related to the process of creation of
electron-positron pairs from the high pressure in the star which results in
the reduction of the photon pressure, consequently an initial collapse and an
explosion. Under certain conditions, the star can survive with lighter mass,
outside the forbidden region, after losing its mass partially by expelling its
outer layers during series of explosions. This is called PPISN (Yoshida et
al., 2016). Conversely, after PISN, no compact remnant is left behind.
Therefore finding a BH with a mass in the forbidden region may indicate having
a higher generation BH which could not have originated directly from the
stellar evolution (Samsing & Hotokezaka, 2020). Besides the higher generation
BH scenario, such cases may also be considered under, but not limited to,
primordial BHs or BHs involving particles beyond the standard model such as
dark matter (Bird et al., 2016; Clesse & García-Bellido, 2017).

However, theoretical and numerical studies show that the bounds are not
precise and also can vary with the initial composition of the star (Woosley,
2017). Besides the limits on the upper mass gap, the shape of the first
generation BH mass distribution is not observationally well constrained and
there are few appropriate mass distributions. However, the most recent
estimates favor a power law+peak model the most (Abbott et al., 2020e). The
peak is thought to carry the BHs which survived a PPISN.

Due to all of these uncertainties, we perform our search using different mass
distributions separately. We adopt a single parametrization which is the power
law+peak model in (Abbott et al., 2020e); but with different upper bounds
$=\\{50,60,70,100\\}M_{\odot}$. Except the upper bound, we use the fitted
parameters for the power law+peak model in Abbott et al. (2020e). We obtain
the second generation BHs’ mass distribution by adding the primary and
secondary masses of the #1 mergers as random variables, using their joint
distribution. Finally, since we are interested in detected mergers, we modify
the mass distributions by multiplying by $m_{secondary}^{1.5}$ which accounts
for the detection bias. This approximate mass factor favoring heavier masses
for detection comes analytically when imposing a threshold on signal-to-noise
ratio (SNR) for detections for the uniformly distributed sources in space.
Although the detection threshold is the false-alarm rate not the SNR in
reality because of the non-gaussian Poisson-like noise called glitches, it is
well correlated with SNR especially after glitch involving data parts are
removed (Abbott et al., 2020a).

###  2.3 The background distribution

Our significance test is based on a frequentist $p$-value assignment via
comparison with a background distribution. The background distribution
involves simulated first generation BH mergers randomly matched with each
other, which mimic the set of unrelated merger pairs. The background
distribution generation algorithm is based on performing BBH merger
simulations and localizing them with the BAYESTAR software package (Singer &
Price, 2016; Singer et al., 2016). BAYESTAR is a fast localization software
for GW detection from compact binary coalescences which takes $\mathcal{O}(1)$
minute to localize a merger by a central processing unit (CPU) core with a
frequency of $\sim 1$ GHz. It was used for the fast-response skymaps for open
public alerts during O3 which were used by multi-messenger searches. It is
slightly less accurate than the localization from the computationally
expensive full parameter estimation (Singer & Price, 2016). However the full
parameter estimation may take several days to finish in the same configuration
per merger and hence is not appropriate for a background distribution
generation. We construct distributions at 2 sensitivities, O1/O2 and O3
sensitivity, and for relevant detector combinations in the runs, considering 2
LIGO (Aasi et al., 2015) (Hanford and Livingston) and the Virgo (Acernese et
al., 2014) detectors.

In order to construct the background distributions for our test statistic, we
need the same inputs as real detections which have parameter estimations with
variances. We model joint distributions of component masses of the second
merger as bivariate normal distributions truncated such that the larger mass
is always greater than the smaller one. The mean location of the distributions
are chosen as the injected actual masses. The elements of the covariance of
the distribution are fitted as a linear function of the detection signal-to-
noise ratio (SNR). The parameters of the fit is extracted from the
distributions of the real detections of GWTC-2. Similarly, we model the
distributions of final mass and $\chi_{\text{eff}}$ of the first merger as
normal distributions, whose variances are found from linear fits of SNR. For
$\chi_{\text{eff}}$ we limit the distributions between [-1,1]. In accordance
with our assumption for BHs involved in unrelated mergers (spin magnitude
uniform in [0,1] and random orientations) we assign a $\chi_{\text{eff}}$
value to each merger. This $\chi_{\text{eff}}$ value and the 5% reduced total
mass of the merger constitute the mean location of the distributions.

Before moving on the results of our search, in order to estimate the possible
capability of our search, we performed simulations for the triple hierarchical
merger scenario and found that if two mergers of such an interaction are
detected by 2 LIGO and the Virgo detectors (3 detector detection) at O3
sensitivity, they can be identified by our search as a triple hierarchical
merger at $\sim 90\%$ efficiency at $3\sigma$ confidence level ($p\lesssim$
1/740).

##  3 Results and Discussion

Here we present the results of our search. Table 1 shows the most significant
event pairs and their individual $p$-values for different upper bounds of the
first generation BHs’ mass distribution. We also provide $p$-values for
partial inputs: time, mass and spin only; and time and volume only. The
$p$-values in Table 1 are for individual events and do not consider the
multiple hypothesis testing correction. We discuss the multiple hypothesis
testing correction at the end of this section.

| Upper bound on 1G BH
---
mass distribution [$M_{\odot}]$
Most significant pairs |  | Individual $p$-value:
---
Overall – Time, mass and spin only – Time and volume only
50 |  | All pairs of GW190521
---
GW190521-GW190514
GW190521-GW170823
GW190521-GW170729
| 0 (null hypothesis is rejected with certainty)
---
0 – 0 – 0.14 (0.14 for eccentric GW190521)
0 – 0 – 0.45 (0.24 for eccentric GW190521)
0 – 0 – 0.32 (0.31 for eccentric GW190521)
60 |  | GW190519-GW170818
---
GW190915-GW190708
GW190910-GW190512
|  $1.1\times 10^{-3}$ – $4.9\times 10^{-2}$ – $3.9\times 10^{-2}$
---
$9.0\times 10^{-3}$ – $0.20$ – $3.2\times 10^{-2}$
$1.3\times 10^{-2}$ – $0.25$ – $1.8\times 10^{-2}$
70 |  | GW190519-GW170818
---
GW190910-GW190512
GW190915-GW190708
|  $1.7\times 10^{-3}$ – $7.7\times 10^{-2}$ – $3.9\times 10^{-2}$
---
$8.1\times 10^{-3}$ – $0.26$ – $1.8\times 10^{-2}$
$8.4\times 10^{-3}$ – $0.19$ – $3.2\times 10^{-2}$
100 |  | GW190519-GW170818
---
GW190910-GW190512
GW190915-GW190708
|  $1.5\times 10^{-3}$ – $5.6\times 10^{-2}$ – $3.9\times 10^{-2}$
---
$7.2\times 10^{-3}$ – $0.21$ – $1.8\times 10^{-2}$
$7.5\times 10^{-3}$ – $0.18$ – $3.2\times 10^{-2}$
Table 1: The most significant pairs and their significances for the triple
hierarchical merger scenario for different upper bounds for the mass
distribution of first generation BHs assuming the power law+peak model. The
given $p$-values do not include the multiple hypothesis correction factor.

We find that for upper bound$=50M_{\odot}$, GW190521 produces an infinite TS.
This is caused by the fact that the mass estimation sample for the heavier
component of GW190521 only involves masses greater than $50M_{\odot}$. For the
eccentric detection of GW190521, the parameters of the component masses for
the best matching template are also well above $50M_{\odot}$ at $102M_{\odot}$
(Gayathri et al., 2020b). Therefore our null hypothesis, in this case, is
rejected with certainty. One of the plausible explanations is it being a
result of a previous merger. We provide the three significant pairs involving
GW190521 with the partial search which uses only the time ordering and volume
localizations: GW190521-GW190514, GW190521-GW170823 and GW190521-GW170729. The
eccentric localization, which has a closer distance reconstruction, produces a
substantially higher significance for GW190521-GW170823 pair. However, the
parameters of the component masses of GW190521 for the best matching eccentric
template is beyond the reach of the final mass of GW170823. Therefore without
a new and proper parameter estimation for the eccentric analysis, GW170823
does not seem to be a plausible predecessor of GW190521 at the moment.
Regarding GW190521-GW190514 and GW190521-GW170729 about the triple
hierarchical merger scenario, there are two points worth mentioning which make
the pairs more notable:

  * •

The final masses of both GW190514 and GW170729 partially agree with the non-
eccentric estimation and the best matching eccentric template’s value of the
heavier mass of GW190521.

  * •

GW190514’s 90% sky localization is in agreement with the location of the AGN
from which a candidate optical counterpart was detected after GW190521 by the
Zwicky Transient Facility (Graham et al., 2020). The distance reconstruction
of the mergers are nearly identical along the AGN’s direction.

For 60, 70 and 100 $M_{\odot}$ for the mass upper bound, GW190519-GW170818
appear as the most significant pair with $\sim 3\sigma$ individual
significance. When we investigate the pair’s properties, we see that there is
not a dominating single input parameter that makes the pair significant; but
all the properties contribute to the significance. This can be seen from the
$p$-values of the partial searches in Table 1. We note that the distance
reconstructions of two events seem to be peaked around different distances
(2.85 and 1.06 Gpc); but with non-zero overlap. With a good 2D sky overlap,
the overall volume overlap becomes non-negligible. Their joint distance
reconstruction is peaked around 1.13 Gpc. The significance of the mass term
comes from the overlap of the heavier mass of GW190519
($m_{1}=64.5^{+11.3}_{-13.2}$) (Abbott et al., 2020b) with the final mass of
GW170818 ($m_{f}=59.4^{+4.9}_{-3.8}$) (Abbott et al., 2019a), as seen in Fig.
2. In Fig. 3 we show the parameter estimation of the spin of the heavy
component of GW190519 and the final spin of GW170818 with the uncertainties
provided in GWTC-1(Abbott et al., 2019a) ($a_{f}=0.67^{+0.07}_{-0.08}$). We
see that the GW190519’s heavy component is expected to be a spinning BH whose
dimentionless spin magnitude agrees with the spin of the final BH of GW170818.
In our search due to limitations mentioned in Sec. 2.1 we do not use
individual spins; but the $\chi_{\text{eff}}$ parameter of the merger #2. This
choice does not directly favor spinning BHs. However, in this case, we do have
a spinning BH which is expected from hierarchical merger scenarios and in
particular the triple hierarchical merger scenario. We show the sky
localizations of two mergers in Fig. 4.

Figure 2: Probability densities for the mass estimations of the heavy
component of GW190519 (blue) and the total mass of GW170818 reduced by 5% for
GW radiation (orange) Figure 3: Probability density distribution of the
dimensionless spin magnitude for the heavy component of GW190519 from its
parameter estimation (blue) and range of it for the final BH of GW170818 with
its uncertainties from GWTC-1(Abbott et al., 2019a) (orange) Figure 4: Sky
localizations of GW190519 and GW170818 in equatorial coordinates. Darker color
represents higher probability density.

We also list GW190915-GW190708 and GW190910-GW190512 as the next two most
significant pairs in Table 1 with significances $p\lesssim 1\%$. The pairs
including GW190521 have relatively lower significances for higher mass upper
bounds due to the relatively heavy secondary mass of GW190521 which does not
fit well to the expected secondary masses for the triple hierarchical merger
scenario. However, if GW190521 is considered a hierarchical merger, its
heavier secondary mass can be explained by higher order hierarchical mergers
(i.e. 2G+2G mergers (Fragione & Kocsis, 2019; Fragione et al., 2020)).

In our test statistic we did not use the eccentricity of the mergers due to
lack of analyses on the eccentricity of the mergers. However in the
hierarchical merger scenario, the first merger especially is expected to have
non-zero eccentricity (Samsing & Ilan, 2019). Romero-Shaw et al. (2019) finds
events in GWTC-1 have low eccentricity. This observation decreases the
plausiblities of GW170818, GW170729 and GW170823 being the first merger of a
triple hierarchical merger chain. For the events in GWTC-2, only GW190521 is
analyzed for eccentricity and it is claimed to be consistent with $>0.1$
eccentricity (Gayathri et al., 2020b; Romero-Shaw et al., 2020). This is
specifically interesting for the GW190521-GW190514 pair, since two mergers
have only one week in between. In such a short time period, the second merger
would not have enough time to circularize and consequently is also expected to
be eccentric. Therefore eccentric characteristics of GW190521 supports the
possibility of GW190521-GW190514 pair to be a triple hierarchical merger
chain. Further analysis of GW190514 can illuminate this claim more.

Finally, although we have individually significant events, it should be noted
that for each upper bound on the mass distribution, we have analyzed 1431
pairs. Therefore a multiple hypothesis testing factor should be included for
the analysis to bound the family-wise error rate, i.e. it is expected to have
an pair with 1% $p$-value if 100 unrelated pairs are analyzed by the
definition of $p$-value. This does not change the interpretation for the upper
bound=$50M_{\odot}$ case. For all the other limits, the search can only show
the interesting merger pairs; but cannot provide a statistically significant
pair with a low overall false alarm rate.

##  4 Conclusion

We presented a search for triple hierarchical mergers. We analyzed the events
published in GWTC-1 (Abbott et al., 2019a), GWTC-2 (Abbott et al., 2020b)
(except the single detector detection GW190424), by the IAS-Princeton group
(Venumadhav et al., 2020; Zackay et al., 2019a, b) and the eccentric
localization of GW190521 (Gayathri et al., 2020b). Due to the uncertainties in
the mass distribution of astrophysical first generation BHs, we considered
four mass distributions which obey the power law+peak model (Abbott et al.,
2020e); but with different upper bounds. Our results demonstrate the
importance of the upper limit of the mass distribution for the inference of
the origins of BHs.

For upper bound=$50M_{\odot}$ case, we find that GW190521 cannot involve two
first generation BHs with certainty. Based on only time ordering and overlap
of volume localization, for the non-eccentric localization we provide the two
most plausible predecessors; GW190514 and GW170729. Both of the final masses
of these mergers agree with the heavier mass of GW190521. In addition, the AGN
from which a candidate optical counterpart was detected after GW190521 (Graham
et al., 2020) is in the 90% sky localization of GW190514. For the eccentric
localization of GW190521, GW170823 has a better volume overlap than GW170729.
However, the parameter estimation of the eccentric localization does not exist
and based on the parameters of the best matching eccentric waveform (component
masses=102 $M_{\odot}$) GW170823 does not seem as a plausible predecessor.
These differences between the circular and eccentric estimations show the
importance of the assumptions used in the detections for deducing the origin
of the mergers. When we increase the upper bound to 60, 70, and 100
$M_{\odot}$, we find the GW190519-GW170818 pair to be most significant with
$\sim 3\sigma$ significance. In addition to good overlap of volume and mass
estimations of the merger pair, the mathcing BH of GW190519 is estimated to be
spinning which is an expected characteristic of the triple hierarchical merger
scenario and of hierarchical mergers in general. Despite the individual
significances of the merger pairs, the whole search does not yield a
statistically significant finding (except 50 $M_{\odot}$ upper bound) after
the multiple hypothesis testing correction. Instead, the search provides
interesting pairs which can be investigated in astrophysical contexts, i.e.
GW190521-GW190514 pair as a hierarchical merger in an AGN (Samsing et al.,
2020; Yang et al., 2019), eccentricity of GW190514 or search for an AGN at the
common localization of GW190519 and GW170818. Finally, in our test statistic
we have not used the eccentricity as a parameter due to the lack of the proper
eccentricity estimations for the mergers. With the development in the
eccentricity analyses with produced waveforms, our search can be made more
powerful by also using the eccentricity information, which is an expected
characteristic in the triple hierarchical merger scenario (Samsing & Ilan,
2019).

## Acknowledgements

The authors thank Stephen Fairhurst for useful feedback. This document was
reviewed by the LIGO Scientific Collaboration under the document number
P2000459.

We acknowledge computing resources from Columbia University’s Shared Research
Computing Facility project, which is supported by NIH Research Facility
Improvement Grant 1G20RR030893-01, and associated funds from the New York
State Empire State Development, Division of Science Technology and Innovation
(NYSTAR) Contract C090171, both awarded April 15, 2010. The authors thank the
University of Florida and Columbia University in the City of New York for
their generous support. The Columbia Experimental Gravity group is grateful
for the generous support of the National Science Foundation under grant
PHY-2012035.

D.V. acknowledges support from Fulbright foreign student program and Jacob
Shaham Fellowship. A.S. is grateful for the support of the Columbia College
Science Research Fellows program. R.B. is supported by the School of Physics
and Astronomy at the University of Birmingham and the Birmingham Institute for
Gravitational Wave Astronomy. J.S. is supported by the European Unions Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No. 844629.

The samples for the IAS-Princeton detections were retrieved from
https://github.com/jroulet/O2_samples.

We thank V. Gayathri et. al. for sharing the eccentric localization of
GW190521.

This research has made use of data, software and/or web tools obtained from
the Gravitational Wave Open Science Center (https://www.gw-openscience.org/),
a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo
Collaboration. LIGO Laboratory and Advanced LIGO are funded by the United
States National Science Foundation (NSF) as well as the Science and Technology
Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS),
and the State of Niedersachsen/Germany for support of the construction of
Advanced LIGO and construction and operation of the GEO600 detector.
Additional support for Advanced LIGO was provided by the Australian Research
Council. Virgo is funded, through the European Gravitational Observatory
(EGO), by the French Centre National de Recherche Scientifique (CNRS), the
Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef,
with contributions by institutions from Belgium, Germany, Greece, Hungary,
Ireland, Japan, Monaco, Poland, Portugal, Spain.

## References

  * Aasi et al. (2015) Aasi, J., Abbott, B. P., Abbott, R., et al. 2015, Classical and Quantum Gravity, 32, 074001, doi: 10.1088/0264-9381/32/7/074001
  * Abbott et al. (2019a) Abbott, B., Abbott, R., Abbott, T., et al. 2019a, Physical Review X, 9, doi: 10.1103/physrevx.9.031040
  * Abbott et al. (2016) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, Physical Review Letters, 116, 061102, doi: 10.1103/PhysRevLett.116.061102
  * Abbott et al. (2019b) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2019b, Phys. Rev. D, 100, 104036, doi: 10.1103/PhysRevD.100.104036
  * Abbott et al. (2019c) Abbott, B. P., et al. 2019c, The Astrophysical Journal, 883, 149, doi: 10.3847/1538-4357/ab3c2d
  * Abbott et al. (2020a) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2020a, Classical and Quantum Gravity, 37, 055002, doi: 10.1088/1361-6382/ab685e
  * Abbott et al. (2020b) Abbott, R., Abbott, T. D., Abraham, S., et al. 2020b.  https://arxiv.org/abs/2010.14527
  * Abbott et al. (2020c) —. 2020c.  https://arxiv.org/abs/2010.14529
  * Abbott et al. (2020d) —. 2020d, ApJ, 900, L13, doi: 10.3847/2041-8213/aba493
  * Abbott et al. (2020e) —. 2020e.  https://arxiv.org/abs/2010.14533
  * Abbott et al. (2021) Abbott, R., Abbott, T., Abraham, S., et al. 2021, SoftwareX, 13, 100658, doi: 10.1016/j.softx.2021.100658
  * Acernese et al. (2014) Acernese, F., Agathos, M., Agatsuma, K., et al. 2014, Classical and Quantum Gravity, 32, 024001, doi: 10.1088/0264-9381/32/2/024001
  * Antonini & Rasio (2016) Antonini, F., & Rasio, F. A. 2016, ApJ, 831, 187, doi: 10.3847/0004-637X/831/2/187
  * Askar et al. (2017) Askar, A., Szkudlarek, M., Gondek-Rosińska, D., Giersz, M., & Bulik, T. 2017, MNRAS, 464, L36, doi: 10.1093/mnrasl/slw177
  * Bae et al. (2014) Bae, Y.-B., Kim, C., & Lee, H. M. 2014, MNRAS, 440, 2714, doi: 10.1093/mnras/stu381
  * Banerjee et al. (2010) Banerjee, S., Baumgardt, H., & Kroupa, P. 2010, MNRAS, 402, 371, doi: 10.1111/j.1365-2966.2009.15880.x
  * Bartos et al. (2017a) Bartos, I., Haiman, Z., Marka, Z., et al. 2017a, Nature Communications, 8, 831, doi: 10.1038/s41467-017-00851-7
  * Bartos et al. (2017b) Bartos, I., Kocsis, B., Haiman, Z., & Márka, S. 2017b, ApJ, 835, 165, doi: 10.3847/1538-4357/835/2/165
  * Belczynski et al. (2016a) Belczynski, K., Holz, D. E., Bulik, T., & O’Shaughnessy, R. 2016a, Nature, 534, 512, doi: 10.1038/nature18322
  * Belczynski et al. (2016b) Belczynski, K., Repetto, S., Holz, D. E., et al. 2016b, ApJ, 819, 108, doi: 10.3847/0004-637X/819/2/108
  * Bellovary et al. (2016) Bellovary, J. M., Mac Low, M.-M., McKernan, B., & Ford, K. E. S. 2016, ApJ, 819, L17, doi: 10.3847/2041-8205/819/2/L17
  * Berti & Volonteri (2008) Berti, E., & Volonteri, M. 2008, The Astrophysical Journal, 684, 822, doi: 10.1086/590379
  * Bird et al. (2016) Bird, S., Cholis, I., Muñoz, J. B., et al. 2016, Physical Review Letters, 116, 201301, doi: 10.1103/PhysRevLett.116.201301
  * Bogdanović et al. (2007) Bogdanović, T., Reynolds, C. S., & Miller, M. C. 2007, The Astrophysical Journal Letters, 661, L147–L150, doi: 10.1086/518769
  * Campanelli et al. (2008) Campanelli, M., Lousto, C. O., & Zlochower, Y. 2008, Phys. Rev. D, 77, 101501, doi: 10.1103/PhysRevD.77.101501
  * Carr et al. (2016) Carr, B., Kühnel, F., & Sandstad, M. 2016, Phys. Rev. D, 94, 083504, doi: 10.1103/PhysRevD.94.083504
  * Cholis et al. (2016) Cholis, I., Kovetz, E. D., Ali-Haïmoud, Y., et al. 2016, Phys. Rev. D, 94, 084013, doi: 10.1103/PhysRevD.94.084013
  * Clesse & García-Bellido (2017) Clesse, S., & García-Bellido, J. 2017, Physics of the Dark Universe, 15, 142–147, doi: 10.1016/j.dark.2016.10.002
  * Corley et al. (2019) Corley, K. R., Bartos, I., Singer, L. P., et al. 2019, MNRAS, 488, 4459, doi: 10.1093/mnras/stz2072
  * De Mink & Mandel (2016) De Mink, S. E., & Mandel, I. 2016, Monthly Notices of the Royal Astronomical Society, 460, 3545, doi: 10.1093/mnras/stw1219
  * Dominik et al. (2013) Dominik, M., Belczynski, K., Fryer, C., et al. 2013, ApJ, 779, 72, doi: 10.1088/0004-637X/779/1/72
  * Dominik et al. (2012) —. 2012, ApJ, 759, 52, doi: 10.1088/0004-637X/759/1/52
  * Dominik et al. (2015) Dominik, M., Berti, E., O’Shaughnessy, R., et al. 2015, ApJ, 806, 263, doi: 10.1088/0004-637X/806/2/263
  * D’Orazio & Loeb (2018) D’Orazio, D. J., & Loeb, A. 2018, Physical Review D, 97, doi: 10.1103/physrevd.97.083008
  * Fragione et al. (2019) Fragione, G., Grishin, E., Leigh, N. W. C., Perets, H. B., & Perna, R. 2019, MNRAS, 488, 47, doi: 10.1093/mnras/stz1651
  * Fragione & Kocsis (2019) Fragione, G., & Kocsis, B. 2019, Monthly Notices of the Royal Astronomical Society, 486, 4781–4789, doi: 10.1093/mnras/stz1175
  * Fragione et al. (2020) Fragione, G., Loeb, A., & Rasio, F. A. 2020, The Astrophysical Journal Letters, 895, L15, doi: 10.3847/2041-8213/ab9093
  * Gayathri et al. (2020a) Gayathri, V., Bartos, I., Haiman, Z., et al. 2020a, The Astrophysical Journal Letters, 890, L20, doi: 10.3847/2041-8213/ab745d
  * Gayathri et al. (2020b) Gayathri, V., Healy, J., Lange, J., et al. 2020b.  https://arxiv.org/abs/2009.05461
  * Graham et al. (2020) Graham, M., Ford, K., McKernan, B., et al. 2020, Physical Review Letters, 124, doi: 10.1103/physrevlett.124.251102
  * Hamers et al. (2018) Hamers, A. S., Bar-Or, B., Petrovich, C., & Antonini, F. 2018, ApJ, 865, 2, doi: 10.3847/1538-4357/aadae2
  * Hoang et al. (2018) Hoang, B.-M., Naoz, S., Kocsis, B., Rasio, F. A., & Dosopoulou, F. 2018, The Astrophysical Journal, 856, 140, doi: 10.3847/1538-4357/aaafce
  * Hong & Lee (2015) Hong, J., & Lee, H. M. 2015, MNRAS, 448, 754, doi: 10.1093/mnras/stv035
  * Janiuk et al. (2017) Janiuk, A., Bejger, M., Charzyński, S., & Sukova, P. 2017, ArXiv e-prints, 51, 7, doi: 10.1016/j.newast.2016.08.002
  * Loeb (2016) Loeb, A. 2016, ApJ, 819, L21, doi: 10.3847/2041-8205/819/2/L21
  * Lousto & Zlochower (2008) Lousto, C. O., & Zlochower, Y. 2008, Phys. Rev. D, 77, 024034, doi: 10.1103/PhysRevD.77.024034
  * Mandel & de Mink (2016) Mandel, I., & de Mink, S. E. 2016, Monthly Notices of the Royal Astronomical Society, 458, 2634, doi: 10.1093/mnras/stw379
  * Marchant, Pablo et al. (2016) Marchant, Pablo, Langer, Norbert, Podsiadlowski, Philipp, Tauris, Thomas M., & Moriya, Takashi J. 2016, A&A, 588, A50, doi: 10.1051/0004-6361/201628133
  * McKernan et al. (2014) McKernan, B., Ford, K. E. S., Kocsis, B., Lyra, W., & Winter, L. M. 2014, MNRAS, 441, 900, doi: 10.1093/mnras/stu553
  * McKernan et al. (2012) McKernan, B., Ford, K. E. S., Lyra, W., & Perets, H. B. 2012, MNRAS, 425, 460, doi: 10.1111/j.1365-2966.2012.21486.x
  * McKernan et al. (2018) McKernan, B., Ford, K. E. S., Bellovary, J., et al. 2018, ApJ, 866, 66, doi: 10.3847/1538-4357/aadae5
  * McKernan et al. (2019) McKernan, B., Ford, K. E. S., Bartos, I., et al. 2019, ApJ, 884, L50, doi: 10.3847/2041-8213/ab4886
  * Murguia-Berthier et al. (2017) Murguia-Berthier, A., MacLeod, M., Ramirez-Ruiz, E., Antoni, A., & Macias, P. 2017, ApJ, 845, 173, doi: 10.3847/1538-4357/aa8140
  * Neyman & Pearson (1933) Neyman, J., & Pearson, E. S. 1933, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 231, 289, doi: 10.1098/rsta.1933.0009
  * Nitz et al. (2019) Nitz, A. H., Capano, C., Nielsen, A. B., et al. 2019, The Astrophysical Journal, 872, 195, doi: 10.3847/1538-4357/ab0108
  * O’Leary et al. (2009) O’Leary, R. M., Kocsis, B., & Loeb, A. 2009, MNRAS, 395, 2127, doi: 10.1111/j.1365-2966.2009.14653.x
  * Park et al. (2017) Park, D., Kim, C., Lee, H. M., Bae, Y.-B., & Belczynski, K. 2017, MNRAS, 469, 4665, doi: 10.1093/mnras/stx1015
  * Portegies Zwart & McMillan (2000) Portegies Zwart, S. F., & McMillan, S. L. W. 2000, ApJ, 528, L17, doi: 10.1086/312422
  * Rodriguez et al. (2018) Rodriguez, C. L., Amaro-Seoane, P., Chatterjee, S., et al. 2018, Phys. Rev. D, 98, 123005, doi: 10.1103/PhysRevD.98.123005
  * Rodriguez & Antonini (2018) Rodriguez, C. L., & Antonini, F. 2018, ApJ, 863, 7, doi: 10.3847/1538-4357/aacea4
  * Rodriguez et al. (2016a) Rodriguez, C. L., Chatterjee, S., & Rasio, F. A. 2016a, Phys. Rev. D, 93, 084029, doi: 10.1103/PhysRevD.93.084029
  * Rodriguez et al. (2016b) Rodriguez, C. L., Haster, C.-J., Chatterjee, S., Kalogera, V., & Rasio, F. A. 2016b, ApJ, 824, L8, doi: 10.3847/2041-8205/824/1/L8
  * Rodriguez et al. (2015) Rodriguez, C. L., Morscher, M., Pattabiraman, B., et al. 2015, Phys. Rev. Lett., 115, 051101, doi: 10.1103/PhysRevLett.115.051101
  * Rodriguez et al. (2016c) Rodriguez, C. L., Zevin, M., Pankow, C., Kalogera, V., & Rasio, F. A. 2016c, ApJ, 832, L2, doi: 10.3847/2041-8205/832/1/L2
  * Romero-Shaw et al. (2020) Romero-Shaw, I., Lasky, P. D., Thrane, E., & Bustillo, J. C. 2020, The Astrophysical Journal Letters, 903, L5, doi: 10.3847/2041-8213/abbe26
  * Romero-Shaw et al. (2019) Romero-Shaw, I. M., Lasky, P. D., & Thrane, E. 2019, Monthly Notices of the Royal Astronomical Society, 490, 5210–5216, doi: 10.1093/mnras/stz2996
  * Samsing (2018) Samsing, J. 2018, Phys. Rev. D, 97, 103014, doi: 10.1103/PhysRevD.97.103014
  * Samsing et al. (2018a) Samsing, J., Askar, A., & Giersz, M. 2018a, ApJ, 855, 124, doi: 10.3847/1538-4357/aaab52
  * Samsing & D’Orazio (2018a) Samsing, J., & D’Orazio, D. J. 2018a, MNRAS, doi: 10.1093/mnras/sty2334
  * Samsing & D’Orazio (2018b) —. 2018b, Monthly Notices of the Royal Astronomical Society, 481, 5445, doi: 10.1093/mnras/sty2334
  * Samsing et al. (2020) Samsing, J., D’Orazio, D. J., Kremer, K., Rodriguez, C. L., & Askar, A. 2020, Physical Review D, 101, doi: 10.1103/physrevd.101.123010
  * Samsing et al. (2019) Samsing, J., Hamers, A. S., & Tyles, J. G. 2019, Phys. Rev. D, 100, 043010, doi: 10.1103/PhysRevD.100.043010
  * Samsing & Hotokezaka (2020) Samsing, J., & Hotokezaka, K. 2020, arXiv e-prints, arXiv:2006.09744.  https://arxiv.org/abs/2006.09744
  * Samsing & Ilan (2018) Samsing, J., & Ilan, T. 2018, Monthly Notices of the Royal Astronomical Society, 476, 1548, doi: 10.1093/mnras/sty197
  * Samsing & Ilan (2019) —. 2019, Monthly Notices of the Royal Astronomical Society, 482, 30, doi: 10.1093/mnras/sty2249
  * Samsing et al. (2014) Samsing, J., MacLeod, M., & Ramirez-Ruiz, E. 2014, ApJ, 784, 71, doi: 10.1088/0004-637X/784/1/71
  * Samsing et al. (2018b) —. 2018b, ApJ, 853, 140, doi: 10.3847/1538-4357/aaa715
  * Samsing & Ramirez-Ruiz (2017) Samsing, J., & Ramirez-Ruiz, E. 2017, ApJ, 840, L14, doi: 10.3847/2041-8213/aa6f0b
  * Samsing et al. (2020) Samsing, J., Bartos, I., D’Orazio, D. J., et al. 2020, arXiv e-prints, arXiv:2010.09765.  https://arxiv.org/abs/2010.09765
  * Sasaki et al. (2016) Sasaki, M., Suyama, T., Tanaka, T., & Yokoyama, S. 2016, Physical Review Letters, 117, 061101, doi: 10.1103/PhysRevLett.117.061101
  * Schrøder et al. (2018) Schrøder, S. L., Batta, A., & Ramirez-Ruiz, E. 2018, ApJ, 862, L3, doi: 10.3847/2041-8213/aacf8d
  * Silsbee & Tremaine (2017) Silsbee, K., & Tremaine, S. 2017, ApJ, 836, 39, doi: 10.3847/1538-4357/aa5729
  * Singer & Price (2016) Singer, L. P., & Price, L. R. 2016, Phys. Rev. D, 93, 024013, doi: 10.1103/PhysRevD.93.024013
  * Singer et al. (2016) Singer, L. P., Chen, H.-Y., Holz, D. E., et al. 2016, The Astrophysical Journal Letters, 829, L15, doi: 10.3847/2041-8205/829/1/l15
  * Stephan et al. (2016) Stephan, A. P., Naoz, S., Ghez, A. M., et al. 2016, MNRAS, 460, 3494, doi: 10.1093/mnras/stw1220
  * Stone et al. (2017) Stone, N. C., Metzger, B. D., & Haiman, Z. 2017, MNRAS, 464, 946, doi: 10.1093/mnras/stw2260
  * Tanikawa (2013) Tanikawa, A. 2013, MNRAS, 435, 1358, doi: 10.1093/mnras/stt1380
  * VanLandingham et al. (2016) VanLandingham, J. H., Miller, M. C., Hamilton, D. P., & Richardson, D. C. 2016, ApJ, 828, 77, doi: 10.3847/0004-637X/828/2/77
  * Venumadhav et al. (2020) Venumadhav, T., Zackay, B., Roulet, J., Dai, L., & Zaldarriaga, M. 2020, Physical Review D, 101, doi: 10.1103/physrevd.101.083030
  * Veske et al. (2020) Veske, D., Márka, Z., Sullivan, A. G., et al. 2020, Monthly Notices of the Royal Astronomical Society: Letters, 498, L46, doi: 10.1093/mnrasl/slaa123
  * Vigna-Gómez et al. (2020) Vigna-Gómez, A., Toonen, S., Ramirez-Ruiz, E., et al. 2020.  https://arxiv.org/abs/2010.13669
  * Vitale et al. (2017) Vitale, S., Lynch, R., Sturani, R., & Graff, P. 2017, Classical and Quantum Gravity, 34, 03LT01, doi: 10.1088/1361-6382/aa552e
  * Woosley (2016) Woosley, S. E. 2016, ApJ, 824, L10, doi: 10.3847/2041-8205/824/1/L10
  * Woosley (2017) Woosley, S. E. 2017, The Astrophysical Journal, 836, 244, doi: 10.3847/1538-4357/836/2/244
  * Yang et al. (2020) Yang, Y., Bartos, I., Haiman, Z., et al. 2020, ApJ, 896, 138, doi: 10.3847/1538-4357/ab91b4
  * Yang et al. (2019) —. 2019, ApJ, 876, 122, doi: 10.3847/1538-4357/ab16e3
  * Yang et al. (2019) Yang, Y., Bartos, I., Gayathri, V., et al. 2019, Phys. Rev. Lett., 123, 181101, doi: 10.1103/PhysRevLett.123.181101
  * Yoshida et al. (2016) Yoshida, T., Umeda, H., Maeda, K., & Ishii, T. 2016, Monthly Notices of the Royal Astronomical Society, 457, 351–361, doi: 10.1093/mnras/stv3002
  * Zackay et al. (2019a) Zackay, B., Dai, L., Venumadhav, T., Roulet, J., & Zaldarriaga, M. 2019a.  https://arxiv.org/abs/1910.09528
  * Zackay et al. (2019b) Zackay, B., Venumadhav, T., Dai, L., Roulet, J., & Zaldarriaga, M. 2019b, Physical Review D, 100, doi: 10.1103/physrevd.100.023007
  * Zevin et al. (2017) Zevin, M., Pankow, C., Rodriguez, C. L., et al. 2017, ApJ, 846, 82, doi: 10.3847/1538-4357/aa8408
  * Zevin et al. (2019) Zevin, M., Samsing, J., Rodriguez, C., Haster, C.-J., & Ramirez-Ruiz, E. 2019, ApJ, 871, 91, doi: 10.3847/1538-4357/aaf6ec