# The second moment of derivatives of $GL(2)$ $L$-functions over quadratic
twists

Sumit Kumar, Kummari Mallesham, Prahlad Sharma and Saurabh Kumar Singh  Sumit
Kumar, Prahlad Sharma
Alfréd Rényi Institute of Mathematics, Budapest, Hungary;
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Kummari Mallesham
Indian Institute of Technology, Bombay, India;
Email<EMAIL_ADDRESS>Saurabh Kumar Singh
Indian Institute of Technology, Kanpur, India;
Email<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be an Hecke eigenform for the group $\Gamma_{0}(q)$ and $\chi_{d}$ be
a primitive quadratic character of conductor $|d|$. In this article, we prove
an asymptotic for the second moment of the derivative of
$L(s,f\otimes\chi_{8d})$ at the central point $1/2$, which was previously
known under GRH by Petrow [8].

##  1\. Introduction

Let $f$ be a Hecke eigenform for the group $\Gamma_{0}(q)$ of even weight $k$
with trivial central character. We write its Fourier expansion as

$f(z)=\sum_{n=1}^{\infty}\lambda_{f}(n)\,n^{(k-1)/2}\,e(nz),\,\,z\in\mathbb{H}.$

with $\lambda_{f}(1)=1$ and $f$ has been normalised so that the Deligne’s
bound gives $|\lambda_{f}(n)|\leq\tau(n)$ for all $n\geq 1$, where $\tau(n)$
is the divisor function. Let $d$ be a fundamental discriminant coprime to $q$,
and let $\chi_{d}(\cdot)=\left(\frac{d}{\cdot}\right)$ denote the primitive
quadratic character of conductor $|d|$. Then $f\otimes\chi_{d}$ is a newform
on $\Gamma_{0}(q|d|^{2})$ and the twisted $L$-function is given by

$L(s,f\otimes\chi_{d})=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\,\chi_{d}(n)}{n^{s}},\,\,\Re(s)>1.$

It has an analytic continuation to the whole of $\mathbb{C}$ and satisfies the
following functional equation

$\Lambda(s,f\otimes\chi_{d})=i^{k}\eta\chi_{d}(-q)\,\Lambda(1-s,f\otimes\chi_{d}),$

where

$\Lambda(s,f\otimes\chi_{d})=\left(\frac{|d|\sqrt{q}}{2\pi}\right)^{s}\Gamma\left(s+\frac{k-1}{2}\right)\,L(s,f\otimes\chi_{d}),$

and $\eta$ is the eigenvalue of the Fricke involution which is always $\pm 1$.
We denote the root number by
$\omega(f\otimes\chi_{d}):=i^{k}\eta\chi_{d}(-q)$. Note that if $d$ is a
fundamental discriminant, then $\chi_{d}(-1)=1$ if $d$ is positive and
$\chi_{d}(-1)=-1$ if $d$ is negative. The derivative of the $L$-function
$L(s,f\times\chi_{d})$ is given by

$L^{\prime}(s,f\otimes\chi_{d})=-\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{d}(n)\log
n}{n^{s}},\,\,\Re(s)>1,$

which also has a functional equation

$\Lambda^{\prime}(s,f\otimes\chi_{d})=-\omega(f\otimes\chi_{d})\,\Lambda^{\prime}(1-s,f\otimes\chi_{d}).$

In this paper, we are interested in the asymptotic behaviour of
$L^{\prime}(s,f\otimes\chi_{d})$ at the central point $1/2$ over the family of
quadratic twists. The mean value of this family has been explored previously
by many authors, particularly by Bump-Friedberg-Hoffstein [1], Murty-Murty
[6], Iwaniec [2], Munshi [4], [5] and Petrow [8]. Petrow, assuming GRH, proved

(1) $\sideset{}{{}^{\star}}{\sum}_{\begin{subarray}{c}(d,2q)=1\\\
\omega(f\otimes\chi_{8d})=-1\end{subarray}}|L^{\prime}\left(1/2,f\times\chi_{8d}\right)|^{2}J\left(\frac{8d}{X}\right)\sim
C_{f}^{\prime}X\log^{3}X,$

where $C_{f}$ is a fixed explicit contant depending on $f$ only and
$J:\mathbb{R}\to\mathbb{R}_{\geq 0}$ is a compactly supported smooth function
having support in $[1/2,2]$. This result was motivated by an important work by
Soundarajan-Young [9], who proved an analog of (1) without the derivative,
i.e.,

(2)
$\sideset{}{{}^{\star}}{\sum}_{\begin{subarray}{c}(d,2q)=1\end{subarray}}|L\left(1/2,f\times\chi_{8d}\right)|^{2}J\left(\frac{8d}{X}\right)\sim
C_{f}X\log X,$

assuming GRH. The above result was proved unconditionally very recently in a
breakthrough work by Li [3]. The main aim of this article is prove (1)
unconditionally.

######  Theorem 1.

Let $f$ and $J$ be as above. Then we have

$\sideset{}{{}^{\star}}{\sum}_{\begin{subarray}{c}(d,2q)=1\\\
\omega(f\otimes\chi_{8d})=-1\end{subarray}}|L^{\prime}\left(1/2,f\times\chi_{8d}\right)|^{2}J\left(\frac{8d}{X}\right)\,=\,C_{f}\,\widetilde{J}(1)\,X\log^{3}X+O\left(X\,(\log
X)^{\frac{5}{2}+\epsilon}\right),$

where $C_{f}$ is some explicit constant depending only on $f$ and
$\widetilde{J}$ is the Mellin transform of $J$.

We combine the methods of Li [3] and Petrow [8] to prove the above theorem. A
similar result holds when $\omega(f\otimes\chi_{8d})=1$.

######  Corollary 1.

We have

$\sideset{}{{}^{\star}}{\sum}_{\begin{subarray}{c}(d,2q)=1\end{subarray}}|L^{\prime}\left(1/2,f\times\chi_{8d}\right)|^{2}J\left(\frac{8d}{X}\right)\,=\,\mathcal{C}_{f}\,\widetilde{J}(1)\,X\log^{3}X+O\left(X\,(\log
X)^{\frac{5}{2}+\epsilon}\right),$

where $\mathcal{C}_{f}$ is some constant depending on $f$ only.

###### Proof.

When $\omega(f\otimes\chi_{8d})=1$, it follows from the functional equation

$\Lambda^{\prime}(s,f\otimes\chi_{8d})=-\,\Lambda^{\prime}(1-s,f\otimes\chi_{8d}),$

that $\Lambda^{\prime}(1/2,f\otimes\chi_{8d})=0$. Thus

$L^{\prime}({1}/{2},f\otimes\chi_{8d})=-L({1}/{2},f\otimes\chi_{8d})\Big{\\{}\log\left(\frac{8|d|\sqrt{q}}{2\pi}\right)\,+\frac{\Gamma^{\prime}\left({k}/{2}\right)}{\Gamma\left({k}/{2}\right)}\Big{\\}}.$

On applying the asymptoic expression

$\sum_{(d,2q)=1}\left(L\left({1}/{2},f\otimes\chi_{8d}\right)\right)^{2}\,J\left(\frac{8d}{X}\right)=C^{\prime}_{f}\,\widetilde{J}(1)\,X\log
X+O\left(X(\log X)^{1/2+\epsilon}\right)$

from Li [3, Theorem 1.1], it follows that

$\displaystyle\sideset{}{{}^{\star}}{\sum}_{\begin{subarray}{c}(d,2q)=1\\\
\omega(f\otimes\chi_{8d})=1\end{subarray}}|L^{\prime}\left(1/2,f\times\chi_{8d}\right)|^{2}J\left(\frac{8d}{X}\right)\,$
$\displaystyle=(\log
X)^{2}\sum_{(d,2q)=1}\left(L\left({1}/{2},f\otimes\chi_{8d}\right)\right)^{2}\,J\left(\frac{8d}{X}\right)+O\left(X\log
X\right)$ $\displaystyle=C_{f}^{\prime}\,\widetilde{J}(1)\,X(\log
X)^{3}+O\left(X(\log X)^{\frac{5}{2}+\epsilon}\right).$

Thus the corollary follows by Theorem 1 and the above expression. ∎

A similar method works for all discriminants $d$ as well. As an application,
we improve upon the lower bound for

$\displaystyle N(X)=\\#\\{0<d\leq X:d\ \mathrm{is\ a\ fundamental\
discriminant\ with}\ (d,q)=1$ $\displaystyle\text{and}\
L^{\prime}(1/2,E\otimes\chi_{d}\neq 0)\\}.$

Here $E$ be a modular elliptic curve over $\mathbb{Q}$ with the $L$-function
$L(s,E)=L(s,f)$ for some eigenform $f$ of weight $2$ and level $q$, where $q$
is the conductor of $E$. Iwaniec [2] proved that
$N(X)\gg_{\epsilon}X^{2/3-\epsilon}$ for any $\epsilon>0$. This bound was
later improved to

(3) $\displaystyle N(X)\gg_{\epsilon}X^{1-\epsilon},$

by Perelli and Pomykala [7]. We improve upon (3) in the following corollary.

######  Corollary 2.

We have

$N(X)\gg X/\log X.$

###### Proof.

Main ingredients to prove Corollary 2 are non-trivial lower bound for the
first moment of $L^{\prime}(1/2,f\otimes\chi_{d})$ and an upper bound for the
second moment of $L^{\prime}(1/2,f\otimes\chi_{d})$ over $d$ (see [7]).
Indeed, by the work of Murty-Murty [6] and Iwaniec [2], we have

(4)
$\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}L^{\prime}({1}/{2},f\otimes\chi_{d})\sim
CX\,\log X,$

for some constant $C\neq 0$, and by Corollary 1 we have

(5)
$M^{\prime}(2):=\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}|L^{\prime}({1}/{2},f\otimes\chi_{d})|^{2}\ll_{f}X\log^{3}X.$

Thus on applying Cauchy’s inequality to the left-hand side of (4), and then
using (5), we have the corollary. ∎

##  2\. Preliminaries

######  Lemma 2.1.

We have

$L^{\prime}(1/2,f\otimes\chi_{d})\,\,\mathbb{I}(\omega(f\otimes\chi_{d})=-1)=\left(1-i^{k}\eta\chi_{d}(-q)\right)\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{d}(n)}{n^{1/2}}\,W\left(\frac{n}{|d|}\right),$

where

$W(y)=\frac{1}{2\pi
i}\int_{(3)}\frac{\Gamma\left(u+{k}/{2}\right)}{\Gamma\left({k}/{2}\right)}\left(\frac{2\pi
y}{\sqrt{q}}\right)^{-u}\,\frac{du}{u^{2}}.$

###### Proof.

See Lemma 3.1 of [8]. ∎

######  Lemma 2.2.

There exists a smooth function $G:\mathbb{R}\to\mathbb{R}_{\geq 0}$ such that

  * •

Support of $G$ lies in $[3/4,2]$

  * •

$G(x)=1$ for all $x\in[1,3/2]$

  * •

$G(x)+G(x/2)=1$ for all $x\in[1,3]$.

###### Proof.

The following function

$G(x)=\begin{cases}0&x\in(-\infty,3/4)\\\
e^{16}\,e^{-1/(x-3/4)^{2}}&x\in[3/4,1)\\\ 1&x\in[1,3/2]\\\
1-G(x/2)&x\in(3/2,2]\\\ 0&x\in(2,\infty)\end{cases}$

is our required one. ∎

Let $H$ be a positive integer. Then the function $F(x)$, defined by

$F(x)=G(x)+G(x/2)+\ldots+G(x/2^{H}),$

satisfies $F(x)=1$ for all $x\in[1,3.2^{H-1}]$ and has the support in
$[3/4,2^{H+1}]$.

######  Remark 1.

  * •

In particular the function

$V(x)=G(2x)+G(x)+G(x/2)$

satisfies $V(x)=1$ for all $x\in[1/2,3]$.

  * •

We have the smooth partition of unity

$\sum_{H=0}^{\infty}G\left(\frac{x}{2^{H}}\right)=1$

for all $x\in[1,\infty)$.

The following proposition is due to Xiannan Li [3, Proposition 3.2].

######  Proposition 1.

Let the smooth function $G(x)$ as above. Then we have

$\sideset{}{{}^{\star}}{\sum}_{M\leq|m|\leq
2M}\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{1/2+it}}\,\left(\frac{m}{n}\right)G\left(\frac{n}{N}\right)\Big{|}^{2}\ll_{f}\left(1+|t|\right)^{2}\left(M+N\,\log\left(2+\frac{N}{M}\right)\right).$

We record the following lemma from Xiannan Li [3, Lemma 6.3].

######  Lemma 2.3.

For any $\mathcal{Y},N\geq 1$, real $t$, positive integer $\ell$, we have

$\sum_{\begin{subarray}{c}(d,2)=1\\\
d\leq\mathcal{Y}\end{subarray}}\Big{|}\sum_{(n,\ell)=1}\frac{\lambda_{f}(n)}{n^{\frac{1}{2}+it}}\,\left(\frac{8d}{n}\right)\,G\left(\frac{n}{N}\right)\Big{|}^{2}\ll\tau(d)^{5}\,\mathcal{Y}\left(1+|t|\right)^{3}\log\left(2+|t|\right),$

where $\tau$ is the divisor function.

We record here the following lemma from Xiannan Li [3, Lemma 2.4]

######  Lemma 2.4.

Let $H:\mathbb{R}_{+}\to\mathbb{R}$ be a Schwartz class function. Let $n$ be
an odd integer. Then we have

$\displaystyle\sum_{(d,2)=1}\left(\frac{8d}{n}\right)\,H\left(\frac{d}{X}\right)$
$\displaystyle=\delta_{\square}(n)\,\frac{X}{2}\,\check{H}(0)\,\prod_{p\mid
n}\left(1-\frac{1}{p}\right)$ $\displaystyle+\frac{X}{2}\sum_{k\neq
0}(-1)^{k}\frac{G_{k}(n)}{n}\,\check{H}\left(\frac{Xk}{2n}\right),$

where $\check{H}$ is Fourier-type transform which defined as

$\check{H}(y)=\int_{-\infty}^{\infty}\left(\cos\left(2\pi xy+\sin\left(2\pi
xy\right)\right)\right)H(x)\,dx.$

Moreover, for $y\neq 0$ we also have

$\check{H}(y)=\frac{1}{2\pi
i}\int_{(1/2)}\tilde{H}(1-s)\,\Gamma(s)\left(\cos+\frac{y}{|y|}\sin\right)\left(\frac{\pi
s}{2}\right)\,\left(2\pi|y|\right)^{-s}\,ds,$

where

$\tilde{H}(s)=\int_{0}^{\infty}H(x)x^{s-1}\,dx$

is the Mellin transform of $H$. And

$G_{k}(n)=\left(\frac{1-i}{2}+\left(\frac{-1}{n}\right)\frac{1+i}{2}\right)\sum_{a\,\rm
mod\,n}\left(\frac{a}{n}\right)\,e\left(\frac{ak}{n}\right)$

is the Gauss like sum.

In the following lemma we record properties of $G_{k}(n)$.

######  Lemma 2.5.

For $m,n$ relatively prime odd integers, we have

$G_{k}(mn)=G_{k}(m)G_{k}(n)$

and for $p^{\alpha}||k$ (set $\alpha=\infty$ for $k=0$), then

$G_{k}(p^{\beta})=\begin{cases}0,\quad&\text{if}\quad\beta\leq\alpha\,\,\,\text{is
odd}\\\ \phi(p^{\beta}),\quad&\text{if}\quad\beta\leq\alpha\,\,\,\text{is
even}\\\ -p^{\alpha},\quad&\text{if}\quad\beta=\alpha+1\,\,\,\text{is even}\\\
\left(\frac{kp^{-\alpha}}{p}\right)p^{\alpha}\,\sqrt{p},\quad&\text{if}\quad\beta=\alpha+1\,\,\,\text{is
odd}\\\ 0,\quad&\text{if}\quad\beta\geq\alpha+2\,\,\,\text{is
odd}.\end{cases}$

##  3\. Proof of Theorem 1

We are interested in getting an asymptotic formula for the smoothed sum

$M^{\prime}_{\text{s}}(2)=\sideset{}{{}^{\star}}{\sum}_{\begin{subarray}{c}(d,2q)=1\\\
\omega(f\otimes\chi_{8d})=-1\end{subarray}}|L^{\prime}\left(1/2,f\otimes\chi_{8d}\right)|^{2}\,J\left(\frac{8d}{X}\right),$

where $J$ is a non-negative smooth function suppored in $[1/2,2]$. Following
Lemma 3.1 of Petrow [8], we express $L^{\prime}(1/2,f\otimes\chi_{8d})$ as a
Dirichlet series using the approximate functional equation

$L^{\prime}(1/2,f\otimes\chi_{8d})\,=\left(1-i^{k}\eta\chi_{8d}(-q)\right)\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{d}(n)}{n^{1/2}}\,W\left(\frac{n}{|8d|}\right),$

which holds if $\omega(f\otimes\chi_{8d})=-1$. Here the cut-off function $W$
is given by

$W(y)=\frac{1}{2\pi
i}\int_{(3)}\frac{\Gamma\left(u+{k}/{2}\right)}{\Gamma\left({k}/{2}\right)}\left(\frac{2\pi
y}{\sqrt{q}}\right)^{-u}\,\frac{du}{u^{2}}.$

Next we split the Dirichlet series into the main part and the tail part. To
this end, we set

$\mathcal{A}(8d):=\mathcal{A}(1/2,f\otimes\chi_{8d})=(1-i^{k}\eta\chi_{8d}(q))\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{1/2}}\,W\left(\frac{n}{\mathcal{M}}\right),$

where $\mathcal{M}=X/(\log X)^{1000}$, and

$\mathcal{B}(8d):=\mathcal{B}(1/2,f\otimes\chi_{8d})=L^{\prime}(1/2,f\otimes\chi_{8d})-\mathcal{A}(1/2,f\otimes\chi_{8d}).$

Theorem 1 follows from the following propositions.

######  Proposition 2.

We have

$\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}\Big{|}\mathcal{B}(1/2,f\otimes\chi_{8d})\Big{|}^{2}\,J\left(\frac{8d}{X}\right)\ll_{f,\epsilon}X(\log
X)^{2}(\log\log X)^{4}.$

######  Proposition 3.

We have

$\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}\Big{|}\mathcal{A}(1/2,f\otimes\chi_{8d})\Big{|}^{2}\,J\left(\frac{8d}{X}\right)=C_{f}\,\check{J}(0)\,X\,(\log
X)^{3}+O\left(X\,(\log X)^{2}\right),$

where $C_{f}$ is a constant depending on $f$.

Indeed, on applying the identity $(a+b)^{2}=a^{2}+b^{2}+2ab$ to
$M^{\prime}_{\text{s}}(2)$ and then Cauchy’s inequality followed by the above
propositions, we get Theorem 1. In the rest of the paper, we prove Proposition
2 (in Section 4) and Proposition 3 (in Section 5).

##  4\. Proof of Proposition 2

Recall that $\mathcal{B}(1/2,f\otimes\chi_{8d})$ is given by

$\left(1-i^{k}\eta\chi_{d}(-q)\right)\,\frac{1}{2\pi
i}\int_{(3)}\frac{(2\pi/\sqrt{q})^{-w}\,\Gamma\left(\frac{k}{2}+w\right)}{\Gamma\left(\frac{k}{2}\right)}\frac{(8d)^{w}-\mathcal{M}^{w}}{w^{2}}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+w}}\,dw.$

By inserting a smooth dyadic partition of unity into the $n$-sum we get

$|\mathcal{B}(1/2,f\otimes\chi_{8d})|\ll\Big{|}\sum_{N\text{-dyadic}}\int_{(3)}\gamma(w)\frac{(8d)^{w}-\mathcal{M}^{w}}{w^{2}}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+w}}G\left(\frac{n}{N}\right)\,dw\Big{|},$

where
$\gamma(w)=\frac{(2\pi/\sqrt{q})^{-w}\,\Gamma\left({k}/{2}+w\right)}{\Gamma\left({k}/{2}\right)}$
and $G$ is as given in Lemma 2.2 and by $N$ dyadic we mean that $N$ takes the
values $2^{H},H=0,1,\cdots$.

Consider the $n$-sum

$\displaystyle\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+w}}G\left(\frac{n}{N}\right)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+w}}V\left(\frac{n}{N}\right)G\left(\frac{n}{N}\right),$

with $V$ as in Remark 1. Using the Mellin inversion we express $G$ as follows:

$G(x)=\frac{1}{2\pi i}\int_{(\epsilon)}\widetilde{G}(s)\,x^{-s}\,ds,$

for any $\epsilon>0$, where $\widetilde{G}(s)$ is the Mellin transform of
$G(x)$ given by

$\widetilde{G}(s)=\int_{0}^{\infty}G(x)\,x^{s-1}dx.$

Therefore the $n$-sum can be expressed as

$\frac{1}{2\pi
i}\int_{(\epsilon)}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+s+w}}V\left(\frac{n}{N}\right)\,N^{s}\,\widetilde{G}(s)\,ds.$

By the change of variable $s\mapsto s-w$ we arrive at

$\frac{1}{2\pi
i}\int_{(\epsilon)}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+s}}V\left(\frac{n}{N}\right)\,N^{s-w}\,\widetilde{G}(s-w)\,ds.$

Next we split the $N$-sum as follows:

$\sum_{N\text{-dyadic}}=\sum_{\begin{subarray}{c}N\text{-dyadic}\\\
N\leq\mathcal{M}\end{subarray}}+\sum_{\begin{subarray}{c}N\text{-dyadic}\\\
\mathcal{M}<N\leq
X\end{subarray}}+\sum_{\begin{subarray}{c}N\,\text{-dyadic}\\\
N>X\end{subarray}}.$

Thus we see that $|\mathcal{B}(1/2,f\otimes\chi_{8d})|^{2}$ is bounded by

$\Big{|}\underbrace{\sum_{\begin{subarray}{c}N\,\text{-dyadic}\\\
N\leq\mathcal{M}\end{subarray}}\mathcal{S}(N,\chi_{8d})}_{\mathcal{B}_{1}(1/2,f\otimes\chi_{8d})}\Big{|}^{2}+\Big{|}\underbrace{\sum_{\begin{subarray}{c}N\text{-dyadic}\\\
\mathcal{M}<N\leq
X\end{subarray}}\mathcal{S}(N,\chi_{8d})}_{\mathcal{B}_{2}(1/2,f\otimes\chi_{8d})}\Big{|}^{2}+\Big{|}\underbrace{\sum_{\begin{subarray}{c}N\text{-dyadic}\\\
N>X\end{subarray}}\mathcal{S}(N,\chi_{8d})}_{\mathcal{B}_{3}(1/2,f\otimes\chi_{8d})}\Big{|}^{2},$

where $\mathcal{S}(N,\chi_{8d})$ is given by

$\frac{1}{2\pi i}\int_{(3)}\gamma(w)\frac{(8d)^{w}-\mathcal{M}^{w}}{w^{2}}\\\
\int_{(\epsilon)}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+s}}V\left(\frac{n}{N}\right)\,N^{s-w}\,\widetilde{G}(s-w)\,ds\,dw.$

We next estimate

$\mathcal{B}_{i}(X):=\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}\,|\mathcal{B}_{i}(1/2,f\otimes\chi_{8d})|^{2}J\left(\frac{8d}{X}\right),$

for $i=1,2,3$ separately. Before embarking this job we record the following
estimates.

$|\Gamma(k/2+w)\,(2\pi/\sqrt{q})^{-w}|\ll(1+|w|)^{-20},\,|\tilde{G}(s)|\ll(1+|s|)^{-20},$

for any $w$ and $s$ having bounded real parts.

###  4.1. Estimates for $\mathcal{B}_{1}(X)$

We move line of integrations $\Re(w)=3$ to $\Re(w)=-1$ and $\Re(s)=\epsilon$
to $\Re(s)=0$, then by Cauchy’s residue theorem we get that

$\displaystyle\mathcal{B}_{1}(1/2,f\otimes\chi_{8d})=\log\left(\frac{8d}{\mathcal{M}}\right)\,\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N\leq\mathcal{M}\end{subarray}}\,\int_{(0)}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+s}}V\left(\frac{n}{N}\right)\,N^{s}\,\tilde{G}(s)\,ds$
$\displaystyle+\Biggl{\\{}\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N\leq\mathcal{M}\end{subarray}}\int_{(-1)}\frac{(2\pi/\sqrt{q})^{-w}\,\Gamma\left(\frac{k}{2}+w\right)}{\Gamma\left(\frac{k}{2}\right)}\frac{(\frac{8d}{N})^{w}-(\frac{\mathcal{M}}{N})^{w}}{w^{2}}$
$\displaystyle\times\frac{1}{2\pi
i}\int_{(0)}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+s}}V\left(\frac{n}{N}\right)\,N^{s}\,\tilde{G}(s-w)\,ds\,dw\Biggr{\\}}.$

Recall that $\mathcal{M}=X/(\log X)^{1000}$ and $X/2\leq 8d\leq 2X$. Note that
the number of dyadic $N$ such that $N\leq\mathcal{M}$ is at most $\log X$. We
conclude that $|\mathcal{B}_{1}(1/2,f\otimes\chi_{8d})|$ is bound above by

$\displaystyle(\log X)(\log\log
X)\sup_{N\leq\mathcal{M}}\int_{-\infty}^{\infty}\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it}}V\left(\frac{n}{N}\right)\Big{|}\,\frac{dt}{(1+|t|)^{20}}$
$\displaystyle+\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N\leq\mathcal{M}\end{subarray}}\frac{N}{\mathcal{M}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it_{1}}}V\left(\frac{n}{N}\right)\Big{|}\frac{dt_{1}}{(1+|t_{2}|)^{20}}\,\frac{dt_{2}}{(1+|t_{1}-t_{2}|)^{20}}.$

By Cauchy’s inequality we get the following bound for
$|\mathcal{B}_{1}(1/2,f\otimes\chi_{8d})|^{2}$

$\displaystyle(\log X)^{2}(\log\log
X)^{2}\sup_{N\leq\mathcal{M}}\int_{-\infty}^{\infty}\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it}}V\left(\frac{n}{N}\right)\Big{|}^{2}\,\frac{dt}{(1+|t|)^{20}}$
$\displaystyle+\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N\leq\mathcal{M}\end{subarray}}\frac{N}{\mathcal{M}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it_{1}}}V\left(\frac{n}{N}\right)\Big{|}^{2}\frac{dt_{1}}{(1+|t_{2}|)^{20}}\,\frac{dt_{2}}{(1+|t_{1}-t_{2}|)^{20}},$

where we have used $\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N\leq\mathcal{M}\end{subarray}}N/\mathcal{M}\ll 1$. Thus on applying
Proposition 1 we see that
$\sideset{}{{}^{\star}}{\sum_{d}}\,|\mathcal{B}_{1}(1/2,f\otimes\chi_{8d})|^{2}J\left(\frac{8d}{X}\right)$
is dominated by

$(\log X)^{2}(\log\log
X)^{2}\sup_{N\leq\mathcal{M}}\left(X+N\log(2+\frac{N}{X})\right)+\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N\leq\mathcal{M}\end{subarray}}\frac{N}{\mathcal{M}}\left(X+N\log(2+\frac{N}{X})\right),$

which is further bounded by $X\,(\log X)^{2}(\log\log X)^{2}.$ Hence we
conclude that

$\sideset{}{{}^{\star}}{\sum}_{d}\,|\mathcal{B}_{1}(1/2,f\otimes\chi_{8d})|^{2}J\left(\frac{8d}{X}\right)\ll
X\,(\log X)^{2}(\log\log X)^{2}.$

###  4.2. Estimates for $\mathcal{B}_{2}(X)$

We move line of integrations $\Re(w)=3$ to $\Re(w)=1/\log X$ and
$\Re(s)=\epsilon$ to $\Re(s)=0$. Then we see that
$|\mathcal{B}_{2}(1/2,f\otimes\chi_{8d})|$ is at most

$\displaystyle\ll$
$\displaystyle\Big{|}\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
\mathcal{M}<N\leq X\end{subarray}}\int_{(1/\log
X)}\int_{(0)}\,\frac{(2\pi/\sqrt{q})^{-w}\,\Gamma\left(\frac{k}{2}+w\right)}{\Gamma\left(\frac{k}{2}\right)}\frac{(\frac{8d}{N})^{w}-(\frac{\mathcal{M}}{N})^{w}}{w^{2}}$
$\displaystyle\times\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+s}}V\left(\frac{n}{N}\right)\,N^{s}\,\tilde{G}(s-w)\,ds\,dw\Big{|}.$

Note that on the line $\Re(w)=1/\log X$ we have the following estimate

$\Big{|}\frac{(8d)^{w}-\mathcal{M}^{w}}{w^{2}}\Big{|}\ll(\log
X)\,\log\left(\frac{8d}{\mathcal{M}}\right)\ll(\log X)(\log\log X).$

Therefore we get the bound

$\displaystyle|\mathcal{B}_{2}(1/2,f\otimes\chi_{8d})|\ll$ $\displaystyle(\log
X)(\log\log X)\,\sum_{\begin{subarray}{c}N-\text{dyadic}\\\ \mathcal{M}<N\leq
X\end{subarray}}\,\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}$
$\displaystyle\times\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it_{1}}}V\left(\frac{n}{N}\right)\Big{|}\,\frac{dt_{1}}{(1+|t_{2}|)^{20}}\,\frac{dt_{2}}{(1+|t_{1}-t_{2}|)^{20}}.$

We apply Cauchy-Scwartz inequality to get

$\displaystyle|\mathcal{B}_{2}(1/2,f\otimes\chi_{8d})|^{2}\ll$
$\displaystyle(\log X)^{2}(\log\log
X)^{3}\,\sum_{\begin{subarray}{c}N-\text{dyadic}\\\ \mathcal{M}<N\leq
X\end{subarray}}\,\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}$
$\displaystyle\times\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it_{1}}}V\left(\frac{n}{N}\right)\Big{|}^{2}\,\frac{dt_{1}}{(1+|t_{2}|)^{20}}\,\frac{dt_{2}}{(1+|t_{1}-t_{2}|)^{20}},$

Where we have used that $\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
\mathcal{M}<N\leq X\end{subarray}}1\ll\log(X/\mathcal{M})\ll\log\log X$. Thus,
we get that the quantity
$\sideset{}{{}^{\star}}{\sum}_{d}\,|\mathcal{B}_{2}(1/2,f\otimes\chi_{8d})|^{2}J\left(\frac{8d}{X}\right)$
is dominated by

$\displaystyle\ll(\log X)^{2}(\log\log
X)^{3}\sum_{\begin{subarray}{c}N-\text{dyadic}\\\ \mathcal{M}<N\leq
X\end{subarray}}\left(X+N\log\left(2+\frac{N}{X}\right)\right)$
$\displaystyle\ll X\,(\log X)^{2}(\log\log X)^{4}.$

We conclude that

$\sideset{}{{}^{\star}}{\sum}_{d}\,|\mathcal{B}_{2}(1/2,f\otimes\chi_{8d})|^{2}J\left(\frac{8d}{X}\right)\ll
X\,(\log X)^{2}(\log\log X)^{4}.$

###  4.3. Estimates for $\mathcal{B}_{3}(X)$

We move line of integrations $\Re(w)=3$ to $\Re(w)=4$ and $\Re(s)=\epsilon$ to
$\Re(s)=0$. Thus we get

$\displaystyle|\mathcal{B}_{3}(1/2,f\otimes\chi_{8d})|^{2}\ll$
$\displaystyle\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N>X\end{subarray}}\left(\frac{X}{N}\right)^{4}\,\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}$
$\displaystyle\times\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{\frac{1}{2}+it_{1}}}V\left(\frac{n}{N}\right)\Big{|}^{2}\,\frac{dt_{1}}{(1+|t_{2}|)^{20}}\,\frac{dt_{2}}{(1+|t_{1}-t_{2}|)^{20}},$

where we have used that $\sum_{\begin{subarray}{c}N\text{-dyadic}\\\
N>X\end{subarray}}\left(\frac{X}{N}\right)^{4}\ll 1$. Thus we see that

$\displaystyle\mathcal{B}_{3}(X)\ll\sum_{\begin{subarray}{c}N-\text{dyadic}\\\
N>X\end{subarray}}\left(\frac{X}{N}\right)^{4}\left(X+N\log\left(2+\frac{N}{X}\right)\right)\ll
X.$

For the second inequality we appeal to the inequality

$\frac{X}{N}\log\left(2+\frac{N}{X}\right)\ll 1$

for $N>X$. We conclude that

$\sideset{}{{}^{\star}}{\sum}_{d}\,|\mathcal{B}_{3}(1/2,f\otimes\chi_{8d})|^{2}J\left(\frac{8d}{X}\right)\ll
X.$

Thus we conclude the proof of Proposition 2.

##  5\.  Proof of Proposition 3

We are seeking an asymptotic formula for

$\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}\left(\mathcal{A}(1/2,f\otimes\chi_{8d})\right)^{2}\,J\left(\frac{8d}{X}\right)=\sum_{(d,2q)=1}\left(\mathcal{A}(1/2,f\otimes\chi_{8d})\right)^{2}\,J\left(\frac{8d}{X}\right)\sum_{a^{2}\mid
d}\mu(a)$

Interchanging the order the summations, we see that

(6)
$\displaystyle\sideset{}{{}^{\star}}{\sum}_{(d,2q)=1}\left(\mathcal{A}(1/2,f\otimes\chi_{8d})\right)^{2}\,J\left(\frac{8d}{X}\right)$
$\displaystyle=2\left(\sum_{\begin{subarray}{c}a\leq Y\\\
(a,2q)=1\end{subarray}}+\sum_{\begin{subarray}{c}a>Y\\\
(a,2q)=1\end{subarray}}\right)\mu(a)\sum_{(d,2q)=1}J\left(\frac{8da^{2}}{X}\right)$
$\displaystyle\times\left(1-i^{k}\eta\chi_{8d}(q)\right)\Bigg{|}\sum_{\begin{subarray}{c}n=1\\\
(n,a)=1\end{subarray}}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{1/2}}\,W\left(\frac{n}{\mathcal{M}}\right)\Bigg{|}^{2},$

where $Y=\log^{200}X$.

###  5.1. Contribution of small $a$, $a\leq Y$

Let us denote the contribution of $a\leq Y$ to (6) by $\mathcal{D(\leq Y)}$.
It is given by

$\displaystyle 2\sum_{\begin{subarray}{c}a\leq Y\\\
(a,2q)=1\end{subarray}}\mu(a)\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{\sqrt{n_{1}n_{2}}}\,W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right)\left(\mathcal{S}_{q^{2}}(a;n_{1}n_{2})-i^{k}\eta\,\mathcal{S}_{q}(a;n_{1}n_{2})\right),$

where

(7)
$\mathcal{S}_{Q}(a;n_{1}n_{1})=\sum_{(d,2)=1}\,\chi_{8d}(n_{1}n_{2}Q)\,J\left(\frac{8da^{2}}{X}\right)$

for $Q\in\\{q,q^{2}\\}$. Using Lemma 2.4 we get

$\displaystyle\mathcal{S}_{Q}(a;n_{1}n_{1})=\delta_{\square}(n_{1}n_{2}Q)\,\frac{X}{16a^{2}}\,\check{J}(0)\,\prod_{p\mid
n_{1}n_{2}Q}\left(1-\frac{1}{p}\right)$
$\displaystyle+\frac{X}{16a^{2}}\sum_{k\neq
0}(-1)^{k}\frac{G_{k}(n_{1}n_{2}Q)}{n_{1}n_{2}Q}\,\check{J}\left(\frac{Xk}{16a^{2}n_{1}n_{2}Q}\right).$

Therefore we have $\mathcal{D(\leq Y)}=M+\mathcal{R}$, where

$\displaystyle M=$
$\displaystyle\frac{X\check{J}(0)}{8}\sum_{Q\in\\{q,q^{2}\\}}\,\varepsilon_{Q}\sum_{\begin{subarray}{c}a\leq
Y\\\
(a,2q)=1\end{subarray}}\frac{\mu(a)}{a^{2}}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
n_{1}n_{2}Q=\square\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1}\lambda_{f}(n_{2}))}{\sqrt{n_{1}n_{2}}}$
$\displaystyle\times\prod_{p\mid
n_{1}n_{2}Q}\left(1-\frac{1}{p}\right)\,W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right),$

and

$\displaystyle\mathcal{R}=\frac{X}{8}\sum_{Q\in\\{q,q^{2}\\}}\,\varepsilon_{Q}\sum_{\begin{subarray}{c}a\leq
Y\\\ (a,2q)=1\end{subarray}}\frac{\mu(a)}{a^{2}}\,T(a,Q)$

with

$\displaystyle T(a,Q)=$ $\displaystyle\sum_{\ell\neq
0}(-1)^{-\ell}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{\sqrt{n_{1}n_{2}}}\,\frac{G_{\ell}(n_{1}n_{2}Q)}{n_{1}n_{2}Q}$
$\displaystyle\times
W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right)\,\check{J}\left(\frac{X\ell}{16a^{2}n_{1}n_{2}Q}\right),$

where $\varepsilon_{q^{2}}=1$ and $\varepsilon_{q}=-i^{k}\eta$. Let

$T(a,Q)=-\left(T_{0}(a,Q)-\sum_{r\geq 1}T_{r}(a,Q)\right)$

where, for $r\geq 0$, $T_{r}(a,Q)$ is defined as

$\displaystyle\sum_{\begin{subarray}{c}\ell\neq 0\\\
\ell=2^{r}\ell^{\prime}\\\
(2,\ell^{\prime})=1\end{subarray}}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{\sqrt{n_{1}n_{2}}}\,\frac{G_{\ell}(n_{1}n_{2}Q)}{n_{1}n_{2}Q}W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right)\,\check{J}\left(\frac{X\ell}{16a^{2}n_{1}n_{2}Q}\right)$
$\displaystyle=\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\underbrace{\sum_{\begin{subarray}{c}(\ell,2)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{\sqrt{n_{1}n_{2}}}\,\frac{G_{2^{\delta(r)}\ell}(n_{1}n_{2}Q)}{n_{1}n_{2}Q}\,\check{J}\left(\frac{X2^{r}\ell}{16a^{2}n_{1}n_{2}Q}\right)}_{\mathbb{T}(n_{1},n_{2})}W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right),$

where $\delta(r)=0$ if $2\mid r$ and $\delta(r)=1$ if $2\nmid r$. For the
second equality we use the fact that $G_{4k}(n)=G_{k}(n)$ for odd $n$.

######  Lemma 5.1.

We have

$\displaystyle T_{r}(a,Q)\ll\,(\log
X)^{2}\,\mathop{\sum\sum}_{N_{1},\,N_{2}\text{-dyadic}}\left(1+\frac{N_{1}}{\mathcal{M}}\right)^{-6}\,\left(1+\frac{N_{2}}{\mathcal{M}}\right)^{-6}I(N_{1},N_{2}),$

where $I(N_{1},N_{2})$ is given by

$\mathop{\int\int\int\int}\Big{|}T(...)\Big{|}\frac{dt_{1}}{\left(1+|t_{1}-t_{3}|\right)^{100}}\frac{dt_{2}}{\left(1+|t_{2}-t_{4}|\right)^{100}}\frac{dt_{3}}{\left(1+|t_{3}|\right)^{100}}\frac{dt_{4}}{\left(1+|t_{4}|\right)^{100}},$

and $T(...)=T(N_{1},N_{2};2^{r}/16a^{2},it_{1},it_{2})$ is defined as in (8).

###### Proof.

We use the dyadic partition of unity to arrive at the following expression of
$T_{r}(a,Q)$

$\mathop{\sum\sum}_{N_{1},N_{2}\text{-dyadic}}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\mathbb{T}(n_{1},n_{2})\,W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right)\,V\left(\frac{n_{1}}{N_{1}}\right)\,V\left(\frac{n_{2}}{N_{2}}\right)\,G\left(\frac{n_{1}}{N_{1}}\right)\,G\left(\frac{n_{2}}{N_{2}}\right),$

which can be further expressed as

$\displaystyle\mathop{\sum\sum}_{N_{1},N_{2}-\,\text{dyadic}}$
$\displaystyle\frac{1}{(2\pi
i)^{2}}\int_{(\sigma_{1})}\int_{(\sigma_{2})}H(w_{1})H(w_{2})\frac{\mathcal{M}^{w_{1}}}{w_{1}^{2}}\frac{\mathcal{M}^{w_{2}}}{w_{2}^{2}}$
$\displaystyle\times\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\mathbb{T}(n_{1},n_{2})}{n_{1}^{w_{1}}n_{2}^{w_{2}}}\,V\left(\frac{n_{1}}{N_{1}}\right)\,V\left(\frac{n_{2}}{N_{2}}\right)\,G\left(\frac{n_{1}}{N_{1}}\right)\,G\left(\frac{n_{2}}{N_{2}}\right)dw_{1}dw_{2},$

where

$H(w)=\frac{\Gamma\left(w+\frac{k}{2}\right)(2/\sqrt{q})^{-w}}{\Gamma(k/2)},$

and $\sigma_{1},\sigma_{2}>0$. On applying the Mellin inversion to the above
expression, we arrive at

$\displaystyle\mathop{\sum\sum}_{N_{1},N_{2}-\,\text{dyadic}}$
$\displaystyle\frac{1}{(2\pi
i)^{4}}\int_{(\sigma_{1})}\int_{(\sigma_{2})}\int_{(2\epsilon)}\int_{(2\epsilon)}H(w_{1})H(w_{2})\tilde{G}(u)\,\tilde{G}(v)\frac{\mathcal{M}^{w_{1}}}{w_{1}^{2}}\frac{\mathcal{M}^{w_{2}}}{w_{2}^{2}}\,N_{1}^{u}N_{2}^{v}$
$\displaystyle\times\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\mathbb{T}(n_{1},n_{2})}{n_{1}^{w_{1}+u}n_{2}^{w_{2}+v}}\,\,du\,dv\,dw_{1}\,dw_{2}.$

We now make the change of variables $u\to u-w_{1}$ and $v\to v-w_{2}$ in the
$u,v$-integrals to arrive at

$\displaystyle\mathop{\sum\sum}_{N_{1},N_{2}-\,\text{dyadic}}$
$\displaystyle\frac{1}{(2\pi
i)^{4}}\int_{(\sigma_{1})}\int_{(\sigma_{2})}\int_{(2\epsilon)}\int_{(2\epsilon)}H(w_{1})H(w_{2})\tilde{G}(u-w_{1})\,\tilde{G}(v-w_{2})$
$\displaystyle\times\frac{(\mathcal{M}/N_{1})^{w_{1}}}{w_{1}^{2}}\frac{(\mathcal{M}/N_{2})^{w_{2}}}{w_{2}^{2}}T(N_{1},N_{2};2^{r}/16a^{2},u,v)\,N_{1}^{u}N_{2}^{v}\,dw_{1}\,dw_{2}\,du\,dv\,.$

where $T(N_{1},N_{2};\alpha,u,v)$ is given by

(8)
$\sum_{\begin{subarray}{c}(\ell,2)=1\end{subarray}}\,\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{n_{1}^{1/2+u}n_{2}^{1/2+v}}\,\frac{G_{2^{\delta(r)}\ell}(n_{1}n_{2}Q)}{n_{1}n_{2}Q}\,\check{J}\left(\frac{\ell
X\alpha}{n_{1}n_{2}Q}\right)V\left(\frac{n_{1}}{N_{1}}\right)V\left(\frac{n_{2}}{N_{2}}\right).$

We now shift the contours in the $w_{1}$ and $w_{2}$ integrals. If
$N_{1}>\mathcal{M}$ and $N_{2}>\mathcal{M}$, we move the contours to
$\Re(\sigma_{1})=6$ and $\Re(\sigma_{2})=6$. If $N_{1}\leq\mathcal{M}$ and
$N_{2}>\mathcal{M}$, we move the contours to $\Re(\sigma_{1})=-6$ and
$\Re(\sigma_{2})=6$, and in this case we get a double pole at $w_{1}=0$ with
the residue

$H^{\prime}(0)\tilde{G}(u)-H(0)\tilde{G}^{\prime}(u)+H(0)\tilde{G}(u)\,\log\left(\frac{\mathcal{M}}{N_{1}}\right).$

The case $N_{1}>\mathcal{M}$ and $N_{2}\leq\mathcal{M}$ is similar. In the
remaining case, i.e., $N_{1}\leq\mathcal{M}$ and $N_{2}\leq\mathcal{M}$ we
move both the contours to $\Re(\sigma_{1})=-6$ and $\Re(\sigma_{2})=-6$ and
encounter double poles at $w_{1}=0$ and $w_{0}=0$. Finally we move both the
$u,v$ integrals to $\Re(u)=0$ and $\Re(v)=0$ and thus we get the lemma by
estimating the $u,v$-integrals trivially. ∎

###  5.2. Estimation of $T(N_{1},N_{2};\alpha,it_{1},it_{2})$

Set $\alpha^{\prime}=\alpha/2$ if $\delta(r)=1$ and $\alpha^{\prime}=\alpha$
if $\delta(r)=0$. Recall that $T(N_{1},N_{2};\alpha^{\prime},it_{1},it_{2})$
is given by

$\sum_{\begin{subarray}{c}(\ell,2)=1\end{subarray}}\,\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{n_{1}^{1/2+it_{1}}n_{2}^{1/2+it_{2}}}\,\frac{G_{2^{\delta(r)}\ell}(n_{1}n_{2}Q)}{n_{1}n_{2}Q}\,\check{J}\left(\frac{\ell
X\alpha^{\prime}}{n_{1}n_{2}Q}\right)\,V\left(\frac{n_{1}}{N_{1}}\right)\,V\left(\frac{n_{2}}{N_{2}}\right).$

Using the expression of $\check{J}$ and the Mellin inversion theorem in the
above expression we arrive at

(9) $\displaystyle\frac{1}{(2\pi
i)^{3}}\int_{(\epsilon)}\int_{(2\epsilon)}\int_{(2\epsilon)}\tilde{V}(u)\tilde{V}(v)\,\tilde{J}(1-s)\Gamma(s)$
$\displaystyle\times\sideset{}{{}^{\star}}{\sum}_{(\ell_{1},2)}\left(\cos+\frac{\ell_{1}}{|\ell_{1}|}\sin\right)\left(\frac{\pi
s}{2}\right)\left(\frac{Q}{|\ell_{1}|X\alpha^{\prime}}\right)^{s}\,$
$\displaystyle\times
Z\left(\frac{1}{2}+it_{1}+u-s,\frac{1}{2}+it_{2}+v-s,s;\ell_{1},a\right)N_{1}^{u}\,N_{2}^{v}\,ds\,du\,dv,$

with

$Z(\alpha,\beta,\gamma;\ell_{1},a)=\sum_{\begin{subarray}{c}\ell_{2}\geq 1\\\
(\ell_{2},2)=1\end{subarray}}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{n_{1}^{\alpha}n_{2}^{\beta}\ell_{2}^{2\gamma}}\,\frac{G_{2^{\delta(r)}\ell_{1}\ell_{2}^{2}}(n_{1}n_{2}Q)}{n_{1}n_{2}Q},$

where we split the sum over $\ell$ as $\ell=\ell_{1}\ell_{2}^{2}$ with
$\ell_{1}$ square free and $\ell_{2}\geq 1$.

The following lemma is similar to Lemma 2.5 of [3](with minor changes).

######  Lemma 5.2.

Let $\ell_{1}$ be square free. Let $m(\ell_{1})=\ell_{1}$ if
$\ell_{1}\equiv\,1\,\rm mod\,4$, and $m(\ell_{1})=4\ell_{1}$ if
$\ell_{1}\,\equiv,\,2,3\rm mod\,\,4$. Then we have

$Z(\alpha,\beta,\gamma;\ell_{1},a)=L\left(1/2+\alpha,f\otimes\chi_{m(\ell_{1})}\right)L\left(1/2+\beta,f\otimes\chi_{m(\ell_{1})}\right)\,Y\left(\alpha,\beta,\gamma;\ell_{1}\right),$

where $Y\left(\alpha,\beta,\gamma;\ell_{1}\right)$ is given by

$\frac{Z_{2}\left(\alpha,\beta,\gamma;\ell_{1},a\right)}{\zeta(1+\alpha+\beta)L\left(1+2\alpha,\text{sym}^{2}f\right)L\left(1+2\beta,\text{sym}^{2}f\right)L\left(1+\alpha+\beta,\text{sym}^{2}f\right)},$

Here $Z_{2}(\alpha,\beta,\gamma;\ell_{1},a)$ is analytic in the region
$\Re(\alpha),\Re(\beta)\geq-\delta/2$ and $\Re(\gamma)\geq 1/2+\delta$ for any
$0<\delta<1/3$. Moreover, in this region we have
$Z_{2}(\alpha,\beta,\gamma;\ell_{1},a)\ll_{f,\delta}\,Q^{-1/2}\,\tau(aQ)$. We
also have the following Dirichlet series represention

$Y\left(\alpha,\beta,\gamma;\ell_{1}\right)=\mathop{\sum\sum\sum}_{r_{1},r_{2},r_{3}}\frac{C(r_{1},r_{2},r_{3})}{r_{1}^{\alpha}r_{2}^{\beta}r_{3}^{2\gamma}},$

for some coefficients $C(r_{1},r_{2},r_{3})$.

###### Proof.

By multiplicativity we have

$Z(\alpha,\beta,\gamma;\ell_{1},a)=\prod_{p}\,\mathcal{F}(p),$

where $\mathcal{F}(2)=1$ and for odd prime $p$, $\mathcal{F}(p)$ is given by

$\mathcal{F}(p)=\begin{cases}\sum_{n_{1},n_{2},\ell_{2}\geq
0}\frac{\lambda_{f}(p^{n_{1}})\lambda_{f}(p^{n_{2}})}{p^{n_{1}\alpha}p^{n_{2}\beta}p^{2\ell_{2}\gamma}}\frac{G_{2^{\delta(r)}\ell_{1}p^{2\ell_{2}}}(p^{n_{1}+n_{2}})}{p^{n_{1}+n_{2}}},\quad\,&\text{if}\,p\nmid
aQ,\\\ \left(1-\frac{1}{p^{2\gamma}}\right)^{-1}\,&\text{if}\,\quad p\mid
a,\,p\nmid Q,\\\ \sum_{n_{1},n_{2},\ell_{2}\geq
0}\frac{\lambda_{f}(p^{n_{1}})\lambda_{f}(p^{n_{2}})}{p^{n_{1}\alpha}p^{n_{2}\beta}p^{2\ell_{2}\gamma}}\frac{G_{2^{\delta(r)}\ell_{1}p^{2\ell_{2}}}(p^{n_{1}+n_{2}+r_{p}})}{p^{n_{1}+n_{2}+r_{p}}},\quad\,&\text{if}\,p\nmid
a,\,\,p^{r_{p}}||Q\\\ \sum_{\ell_{2}\geq
0}\frac{1}{p^{2\ell_{2}\gamma}}\frac{G_{2^{\delta(r)}\ell_{1}p^{2\ell_{2}}}(p^{r_{p}})}{p^{r_{p}}},\quad\,&\text{if
}\,p\mid(a,Q),\,\,p^{r_{p}}||Q.\par\end{cases}$

Let us assume that $\Re(\alpha),\Re(\beta)\geq-c$ and $\Re(\gamma)\geq
1/2+\delta$ for some $0<c<\delta<1/3$. For $p\nmid 2\ell_{1}aQ$, by Lemma 2.5
we have

$\displaystyle\mathcal{F}(p)$ $\displaystyle=\sum_{\ell_{2}\geq
0}\frac{1}{p^{2\ell_{2}\gamma}}\left(\sum_{h=0}^{\ell_{2}}\frac{\phi(p^{2h})}{p^{2h}}\sum_{\begin{subarray}{c}i,j\\\
i+j=2h\end{subarray}}\frac{\lambda_{f}(p^{i})+\lambda_{f}(p^{j})}{p^{i\alpha+j\beta}}+\frac{\chi_{\ell_{1}}(p)}{p^{1/2}}\left(\sum_{\begin{subarray}{c}i,j\\\
i+j=2\ell_{2}+1\end{subarray}}\frac{\lambda_{f}(p^{i})+\lambda_{f}(p^{j})}{p^{i\alpha+j\beta}}\right)\right)$
$\displaystyle=1+\frac{\lambda_{f}(p)\chi_{\ell_{1}}(p)}{p^{1/2}}\left(\frac{1}{p^{\alpha}}+\frac{1}{p^{\beta}}\right)+O\left(\frac{1}{p^{1+2\delta-2c}}\right)$
$\displaystyle=\left(1-\frac{\lambda_{f}(p)\chi_{\ell_{1}}(p)}{p^{1/2+\alpha}}+\frac{\chi_{\ell_{1}}^{2}(p)}{p^{1+2\alpha}}\right)^{-1}\left(1-\frac{\lambda_{f}(p)\chi_{\ell_{1}}(p)}{p^{1/2+\alpha}}+\frac{\chi_{\ell_{1}}^{2}(p)}{p^{1+2\alpha}}\right)^{-1}$
$\displaystyle\times\left(1-\frac{A(p)}{p}\left(\frac{1}{p^{2\alpha}}+\frac{1}{p^{2\beta}}+\frac{1}{p^{\alpha+\beta}}\right)-\frac{1}{p^{1+\alpha+\beta}}+O\left(\frac{1}{p^{3/2-3c}}\right)\right)+O\left(\frac{1}{p^{1+2\delta-2c}}\right),$

and for $p\mid\ell_{1}$ and $p\nmid 2aQ$, we have

$\mathcal{F}(p)=\left(1-\frac{A(p)}{p}\left(\frac{1}{p^{2\alpha}}+\frac{1}{p^{2\beta}}+\frac{1}{p^{\alpha+\beta}}\right)-\frac{1}{p^{1+\alpha+\beta}}+O\left(\frac{1}{p^{3/2-3c}}\right)\right)+O\left(\frac{1}{p^{1+2\delta-2c}}\right),$

where $A(n)$ are deifned by

$L(s,\text{sym}^{2}{f})=\sum_{n}\frac{A(n)}{n^{s}},\quad\Re(s)>1.$

Choose $c\leq\delta/2<1/6$. Note that $\mathcal{F}(p)\ll p^{-1/2}$ for $p\mid
Q$ and

$\prod_{p\mid 2aQ}\left(1+\frac{C}{p^{1/2-c}}\right)\ll
2^{\omega(2aQ)}\ll\tau(aQ),$

where $\omega(n)$ is the number of prime factors of $n$ and $C$ is any
constant. ∎

The following lemma is taken from Li [3, Lemma 5.7].

######  Lemma 5.3.

Suppose $\Re(s)\geq 3/5$ and write $u=-1/2+i\mu,v=-1/2+i\nu$ for real
$\mu,\nu$. We have

$\displaystyle\Big{|}$
$\displaystyle\mathop{\sum}_{r_{1},r_{2},r_{3}}\frac{C(r_{1},r_{2},r_{3})}{r_{1}^{1/2+u+it_{1}}r_{2}^{1/2+v+it_{2}}r_{3}^{2s}}G\left(\frac{r_{1}}{R_{1}}\right)G\left(\frac{r_{2}}{R_{2}}\right)\Big{|}$
$\displaystyle\ll\left(1+|t_{1}|\right)^{1/10}\,\left(1+|t_{2}|\right)^{1/10}(1+|\mu|)(1+|\nu|)\,\exp\left(-c_{1}\sqrt{\log(R_{1}R_{2})}\right),$

where the implied constant and $c_{1}$ may depend on $f$.

######  Remark 2.

In fact, Li [3] proved this lemma with $t_{1}=t,t_{2}=-t$, however his method
also proves the above lemma for any $t_{1}$ and $t_{2}$.

In view of Lemma 5.2 the $u,v$-integrals in (9) are given by

$\displaystyle\sum_{r_{1},r_{2},r_{3}}\frac{C(r_{1},r_{2},r_{3})}{r_{1}^{1/2+it_{1}-s}r_{2}^{1/2+it_{2}-s}r_{3}^{2s}}\mathop{\sum\sum}_{n_{1},n_{2}}\frac{\lambda_{f}(n_{1})\chi_{m(\ell_{1})}(n_{1})}{n_{1}^{1+it_{1}-s}}\frac{\lambda_{f}(n_{2})\chi_{m(\ell_{1})}(n_{2})}{n_{2}^{1+it_{1}-s}}\,V\left(\frac{r_{1}n_{1}}{N_{1}}\right)V\left(\frac{r_{2}n_{2}}{N_{1}}\right)$
$\displaystyle=\mathop{\sum\sum}_{R_{1},R_{2}\,\text{dyadic}}\sum_{r_{1},r_{2},r_{3}}\frac{C(r_{1},r_{2},r_{3})}{r_{1}^{1/2+it_{1}-s}r_{2}^{1/2+it_{2}-s}r_{3}^{2s}}G\left(\frac{r_{1}}{R_{1}}\right)\,G\left(\frac{r_{2}}{R_{2}}\right)$
$\displaystyle\times\mathop{\sum\sum}_{n_{1},n_{2}}\frac{\lambda_{f}(n_{1})\chi_{m(\ell_{1})}(n_{1})}{n_{1}^{1+it_{1}-s}}\frac{\lambda_{f}(n_{2})\chi_{m(\ell_{1})}(n_{2})}{n_{2}^{1+it_{1}-s}}\,V_{1}\left(\frac{n_{1}R_{1}}{N_{1}}\right)\,V_{1}\left(\frac{n_{2}R_{2}}{N_{1}}\right)\,V\left(\frac{r_{1}n_{1}}{N_{1}}\right)V\left(\frac{r_{2}n_{2}}{N_{1}}\right),$

where

$V_{1}(x)=G(4x)+G(2x)+G(x)+G(2x)+G(4x)$

which satisfies $V_{1}(x)=1$ if $x\in[1/4,6]$. We now apply Mellin inversion
theorem to separate variables $r_{i}$ and $n_{i}$ and make some change of
variables to get the following expression for
$T(N_{1},N_{2};\alpha,it_{1},it_{2})$

(10) $\displaystyle\frac{1}{(2\pi
i)^{3}}\int_{(\epsilon)}\int_{(2\epsilon)}\int_{(2\epsilon)}\,\tilde{J}(1-s)\Gamma(s)\,\sideset{}{{}^{\star}}{\sum}_{(\ell_{1},2)}\left(\cos+\frac{\ell_{1}}{|\ell_{1}|}\sin\right)\left(\frac{\pi
s}{2}\right)\left(\frac{QN_{1}N_{2}}{|\ell_{1}|X\alpha^{\prime}}\right)^{s}\tilde{V}(u+s)$
$\displaystyle\times\tilde{V}(v+s)\mathop{\sum\sum}_{R_{1},R_{2}\,\text{dyadic}}\sum_{r_{1},r_{2},r_{3}}\frac{C(r_{1},r_{2},r_{3})}{r_{1}^{1/2+it_{1}+u}r_{2}^{1/2+it_{2}+v}r_{3}^{2s}}G\left(\frac{r_{1}}{R_{1}}\right)\,G\left(\frac{r_{2}}{R_{2}}\right)$
$\displaystyle\times\mathop{\sum\sum}_{n_{1},n_{2}}\frac{\lambda_{f}(n_{1})\chi_{m(\ell_{1})}(n_{1})}{n_{1}^{1+it_{1}+u}}\frac{\lambda_{f}(n_{2})\chi_{m(\ell_{1})}(n_{2})}{n_{2}^{1+it_{1}+v}}\,V_{1}\left(\frac{n_{1}R_{1}}{N_{1}}\right)\,V_{1}\left(\frac{n_{2}R_{2}}{N_{1}}\right)\,N_{1}^{u}N_{2}^{v}\,du\,dv\,ds.$

In the following lemma we give an estimate for
$T(N_{1},N_{2};\alpha,it_{1},it_{2})$.

######  Lemma 5.4.

We have

$\Big{|}T(N_{1},N_{2};\alpha,it_{1},it_{2})\Big{|}\ll\frac{Q\sqrt{N_{1}N_{2}}}{\alpha^{\prime}X}(1+|t_{1}|)^{\frac{17}{10}}\,(1+|t_{1}|)^{\frac{17}{10}}.$

###### Proof.

We split the $\ell_{1}$-sum as follows

$T(...)=T_{(|\ell_{1}|\leq
N_{1}N_{2}/X\alpha^{\prime})}(...)+T_{(|\ell_{1}|>N_{1}N_{2}/X\alpha^{\prime})}(...).$

First we consider the contribution of
$|\ell_{1}|\leq\frac{QN_{1}N_{2}}{X\alpha^{\prime}}$. In this case, in the
expression we move $s,u,v$ contours in (10) to $\Re(s)=3/5$ and
$\Re(u),\Re(v)=-1/2$. Then we estimate the resulting expression using

$|\Gamma(s)\cos s|+|\Gamma(s)\sin s|\ll|s|^{\Re(s)-1/2}$

and Lemma 5.3 to get the following bound for $T_{(|\ell_{1}|\leq
N_{1}N_{2}/X\alpha^{\prime})}(...)$

$\displaystyle\left(\frac{\alpha^{\prime}X}{Q}\right)^{-3/5}(N_{1}N_{2})^{1/10}(1+|t_{1}|)^{1/10}(1+|t_{2}|)^{1/10}\sum_{R_{1},R_{2}\,\text{dyadic}}\exp(-c_{1}\sqrt{\log(R_{1}R_{2})})$
$\displaystyle\times\mathop{\int\int}_{-\infty}^{\infty}\frac{1}{(1+|\mu|)^{10}(1+|\nu|)^{10}}\sum_{|\ell_{1}|\leq\frac{QN_{1}N_{2}}{X\alpha^{\prime}}}\frac{1}{|\ell_{1}|^{3/5}}\Big{|}\sum_{n_{1}}\frac{\lambda_{f}(n_{1})\chi_{m(\ell_{1})}(n_{1})}{n_{1}^{1/2+i\mu+it_{1}}}G\left(\frac{n_{1}}{N_{1}^{\prime}}\right)$
$\displaystyle\times\sum_{n_{2}}\frac{\lambda_{f}(n_{2})\chi_{m(\ell_{1})}(n_{2})}{n_{2}^{1/2+i\nu+it_{1}}}G\left(\frac{n_{2}}{N_{2}^{\prime}}\right)\Big{|}\,d\mu\,d\nu,$

where $N_{1}^{\prime}\asymp N_{1}/R_{1}$ and $N_{2}^{\prime}\asymp
N_{2}/R_{2}$. We recall Lemma 5.3 from [3] which asserts that

(11) $\sideset{}{{}^{\star}}{\sum}_{M\leq|m|\leq
2M}\Big{|}\sum_{n}\frac{\lambda_{f}(n)\chi_{m}(n)}{n^{1/2+it}}\,G\left(\frac{n}{N}\right)\Big{|}^{2}\ll_{f}(1+|t|)^{3}\,M\log(2+|t|).$

We now make a dyadic decomposition in $\ell_{1}$ sum and apply Cauchy-Scwartz
inequality and then by the above estimate we see thay the $\ell_{1}$-sum is
bounded by

$\left(\frac{QN_{1}N_{2}}{X\alpha^{\prime}}\right)^{2/5}(1+|t_{1}|)^{16/10}(1+|t_{2}|)^{16/10}(1+|\mu|)^{1/10}(1+|\nu|)^{1/10}.$

Therefore

$\displaystyle T_{(|\ell_{1}|\leq
N_{1}N_{2}/X\alpha^{\prime})}(...)\ll\frac{Q\sqrt{N_{1}N_{2}}}{\alpha^{\prime}X}(1+|t_{1}|)^{\frac{17}{10}}\,(1+|t_{1}|)^{\frac{17}{10}}.$

In the second case when $|\ell_{1}|>\frac{QN_{1}N_{2}}{\alpha^{\prime}X}$ we
move $s$ integral contour to $\Re(s)=6/5$ and $u,v$ integrals to
$\Re(u),\Re(v)=-1/2$. Therefore we see that
$T_{(|\ell_{1}|>N_{1}N_{2}/X\alpha^{\prime})}(...)$ is dominated by

$\displaystyle\ll\left(\frac{Q}{\alpha^{\prime}X}\right)^{6/5}(N_{1}N_{2})^{7/10}(1+|t_{1}|)^{1/10}(1+|t_{2}|)^{1/10}\sum_{R_{1},R_{2}\,\text{dyadic}}\exp(-c_{1}\sqrt{\log(R_{1}R_{2})})$
$\displaystyle\times\mathop{\int\int}_{-\infty}^{\infty}\frac{1}{(1+|\mu|)^{10}(1+|\nu|)^{10}}\sideset{}{{}^{\star}}{\sum}_{|\ell_{1}|>\frac{QN_{1}N_{2}}{X\alpha^{\prime}}}\frac{1}{|\ell_{1}|^{6/5}}\Big{|}\sum_{n_{1}}\frac{\lambda_{f}(n_{1})\chi_{m(\ell_{1})}(n_{1})}{n_{1}^{1/2+i\mu+it_{1}}}G\left(\frac{n_{1}}{N_{1}^{\prime}}\right)$
$\displaystyle\times\sum_{n_{2}}\frac{\lambda_{f}(n_{2})\chi_{m(\ell_{1})}(n_{2})}{n_{2}^{1/2+i\nu+it_{1}}}G\left(\frac{n_{2}}{N_{2}^{\prime}}\right)\Big{|}\,d\mu\,d\nu.$

Consider $\ell_{1}$ sum

$\sum_{\begin{subarray}{c}J\,\text{dyadic}\\\
J>\frac{QN_{1}N_{2}}{\alpha^{\prime}X}\end{subarray}}\frac{1}{J^{6/5}}\sideset{}{{}^{\star}}{\sum}_{J\leq|\ell_{1}|\leq
2J}\Big{|}\sum_{n_{1}}\frac{\lambda_{f}(n_{1})\chi_{m(\ell_{1})}(n_{1})}{n_{1}^{1/2+i\mu+it_{1}}}G\left(\frac{n_{1}}{N_{1}^{\prime}}\right)\,\sum_{n_{2}}\frac{\lambda_{f}(n_{2})\chi_{m(\ell_{1})}(n_{2})}{n_{2}^{1/2+i\nu+it_{1}}}G\left(\frac{n_{2}}{N_{2}^{\prime}}\right)\Big{|}.$

By Cauchy-Scwartz and (11) we see that the above expression is dominated by

$\displaystyle(1+|t_{1}|)^{16/10}(1+|t_{2}|)^{16/10}(1+|\mu|)^{1/10}(1+|\nu|)^{1/10}\,\sum_{\begin{subarray}{c}J\,\text{dyadic}\\\
J>(QN_{1}N_{2})/\alpha^{\prime}X\end{subarray}}\frac{1}{J^{1/5}}$
$\displaystyle\ll\left(\frac{QN_{1}N_{2}}{\alpha^{\prime}X}\right)^{-1/5}(1+|t_{1}|)^{16/10}(1+|t_{2}|)^{16/10}(1+|\mu|)^{1/10}(1+|\nu|)^{1/10}.$

Therefore we have

$\displaystyle
T_{(|\ell_{1}|>N_{1}N_{2}/X\alpha^{\prime})}(...)\ll\frac{Q\sqrt{N_{1}N_{2}}}{\alpha^{\prime}X}(1+|t_{1}|)^{\frac{17}{10}}\,(1+|t_{1}|)^{\frac{17}{10}}.$

This concludes the proof of the lemma. ∎

We have the following lemma which we gives estimates for $T_{r}(a,Q)$.

######  Lemma 5.5.

We have

$T_{r}(a,Q)\ll\frac{Qa^{2}\mathcal{M}}{2^{r}X}(\log X)^{2}.$

###### Proof.

From Lemma 5.4 and Lemma 5.1 we see that

$\displaystyle T_{r}(a,Q)$ $\displaystyle\ll\frac{Qa^{2}(\log
X)^{2}}{2^{r}X}\mathop{\sum\sum}_{N_{1},N_{2}-\,\text{dyadic}}\left(1+\frac{N_{1}}{\mathcal{M}}\right)^{-6}\,\left(1+\frac{N_{2}}{\mathcal{M}}\right)^{-6}\sqrt{N_{1}N_{2}}$
$\displaystyle\ll\frac{Qa^{2}\mathcal{M}(\log X)^{2}}{2^{r}X}.$

This proves the lemma. ∎

###  5.3. Estimation of $\mathcal{R}$

Recall that

$\displaystyle\mathcal{R}=\frac{X}{8}\sum_{Q\in\\{q,q^{2}\\}}\,\varepsilon_{Q}\sum_{\begin{subarray}{c}a\leq
Y\\\ (a,2q)=1\end{subarray}}\frac{\mu(a)}{a^{2}}\,T(a,Q)$

with

$T(a,Q)=-\left(T_{0}(a,Q)-\sum_{r\geq 1}T_{r}(a,Q)\right).$

By Lemma 5.5 we infer that

$T(a,Q)\ll\frac{Qa^{2}\mathcal{M}}{X}(\log X)^{2}.$

Thus we have

(12) $\mathcal{R}\ll Y\mathcal{M}\ll\frac{X}{(\log X)^{10}}.$

###  5.4. The main term $M$

Recall that

$\displaystyle M=$
$\displaystyle\frac{X\check{J}(0)}{8}\sum_{Q\in\\{q,q^{2}\\}}\,\varepsilon_{Q}\sum_{\begin{subarray}{c}a\leq
Y\\\
(a,2q)=1\end{subarray}}\frac{\mu(a)}{a^{2}}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
n_{1}n_{2}Q=\square\\\
(n_{1}n_{2},2a)=1\end{subarray}}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{\sqrt{n_{1}n_{2}}}$
$\displaystyle\times\prod_{p\mid
n_{1}n_{2}Q}\left(1-\frac{1}{p}\right)\,W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right).$

Upon using

$\sum_{\begin{subarray}{c}a\leq Y\\\
(a,2n_{1}n_{2})=1\end{subarray}}\frac{\mu(a)}{a^{2}}=\frac{8}{\pi^{2}}\prod_{p\mid
n_{1}n_{2}}\left(1-\frac{1}{p^{2}}\right)^{-1}+O\left(\frac{1}{Y}\right),$

we infer that

$\displaystyle M$
$\displaystyle=\frac{X\check{J}(0)}{\pi^{2}}\sum_{Q\in\\{q,q^{2}\\}}\,\varepsilon_{Q}\mathop{\sum\sum}_{\begin{subarray}{c}n_{1},n_{2}\\\
n_{1}n_{2}Q=\square\\\
(n_{1}n_{2},2)=1\end{subarray}}\frac{\lambda_{f}(n_{1}\lambda_{f}(n_{2}))}{\sqrt{n_{1}n_{2}}}\,\prod_{p\mid
n_{1}n_{2}Q}\left(1-\frac{1}{p+1}\right)\,W\left(\frac{n_{1}}{\mathcal{M}}\right)W\left(\frac{n_{2}}{\mathcal{M}}\right)$
$\displaystyle+O\left(\frac{X}{Y}\sum_{\begin{subarray}{c}n_{1}n_{2}=\square\end{subarray}}\frac{\tau(n_{1})\tau(n_{2})}{\sqrt{n_{1}n_{2}}}|W(n_{1}/\mathcal{M})W(n_{2}/\mathcal{M})|\right).$

Note that the above $O$-term is $\frac{X}{Y}(\log X)^{12}$. Following
calculations of Petrow (see [8, p. 1586-1588], we obtain that

$M=C_{f}\,\check{J}(0)\,X\,(\log X)^{3}+O\left(X\,(\log X)^{2}\right),$

where $C_{f}$ is a constant depending only on $f$ and $C_{f}=0$ if and only if
$i^{k}\eta=1$ and $q$ is square. Therefore we have

(13) $D(\leq\mathcal{Y})=C_{f}\,\check{J}(0)\,X\,(\log X)^{3}+O\left(X\,(\log
X)^{2}\right).$

###  5.5. Contribution of large $a$, $a>Y$:

Note that

$\displaystyle\sum_{\begin{subarray}{c}n=1\\\
(n,a)=1\end{subarray}}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{1/2}}\,W\left(\frac{n}{\mathcal{M}}\right)$
$\displaystyle=\sum_{N\,\text{dyadic}}\frac{1}{(2\pi
i)^{2}}\int_{(2)}\frac{\Gamma\left(\frac{k}{2}+w\right)}{(2\pi/\sqrt{q})^{w}\Gamma(\frac{k}{2})}\frac{\mathcal{M}^{w}}{w^{2}}\int_{(\epsilon)}$
$\displaystyle\times\sum_{(n,a)=1}\frac{\lambda_{f}(n)}{n^{1/2+u}}\left(\frac{8d}{n}\right)V\left(\frac{n}{N}\right)N^{u-w}\,\tilde{G}(u-w)\,du\,dw.$

Using Lemma 2.3 and calculations similar to Section 4 we infer that

$\sum_{(d,2)=1}J\left(\frac{8da^{2}}{X}\right)\,\Big{|}\sum_{\begin{subarray}{c}n=1\\\
(n,a)=1\end{subarray}}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{1/2}}\,W\left(\frac{n}{\mathcal{M}}\right)\Big{|}^{2}\,\ll\frac{X(\log
X)^{2}}{a^{2}}\,\tau(a)^{5}.$

Therefore

$\sum_{\begin{subarray}{c}a>Y\\\
(a,2)=1\end{subarray}}\sum_{(d,2)=1}J\left(\frac{8da^{2}}{X}\right)\,\Big{|}\sum_{\begin{subarray}{c}n=1\\\
(n,a)=1\end{subarray}}^{\infty}\frac{\lambda_{f}(n)\chi_{8d}(n)}{n^{1/2}}\,W\left(\frac{n}{\mathcal{M}}\right)\Big{|}^{2}\,\ll\frac{X(\log
X)^{2}}{Y^{1-\epsilon}},$

for any $\epsilon>0$. This is acceptable as we have $Y=(\log X)^{200}$.
Therefore by inserting this expression in (6) we have Proposition 3.

## Acknowledgements

Authors are grateful to Xiannan Li for his encourgement and his interest in
this work. Authors are also thankful to Ritabrata Munshi, Satadal Ganguly and
Surya Ramana for their support. The first and the third author thank Alfréd
Rényi Institute of Mathematics for providing an excellent research
environment.

## References

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  * [2] H. Iwaniec, _On the order of vanishing of modular L-functions at the critical point_ , Séminaire Théorie de Nombres de Bordeaux 2, no. 2 (1990), 365–376.
  * [3] X. Li, _Moments of quadratic twists of modular $L$-functions_, arXiv:2208.07343.
  * [4] R. Munshi, _On mean values and non-vanishing of derivatives of $L$-functions in a nonlinear family_, Compositio Mathematica 147, no. 1 (2011), 19–34.
  * [5] M. Munshi, _On quadratic families of CM elliptic curves_ , Transactions of the AMS 363, no. 8 (2011), 4337–4358.
  * [6] M. R. Murty and V. K. Murty, _Mean values of derivatives of modular $L$\- serie_, Ann. of Math. 133, 447–475(1991).
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  * [8] I. Petrow, _Moments of $L^{\prime}(\frac{1}{2})$ in the family if quadratic twists_, IMRN no. 6, 1576–1612 (2014).
  * [9] K. Soundararajan, and M. P. Young, _The second moment of quadratic twists of modular $L$-functions_, Journal of the European Mathematical Society 12, no. 5 (2010), 1097–1116.


\bar{v}_{1}=(1-\lambda)\bar{v}_{0}+\lambda\bar{v}_{2},\quad
t\in(t_{1},t_{2}),\\\ \bar{v}_{2},\quad t\in(t_{2},\infty),\end{cases}$

where $\lambda=\frac{t-t_{1}}{t_{2}-t_{1}}$.

Assume the minimum realization of $\bar{v}_{0}$ is $v_{0}\in
V^{r}({\mathbb{R}}^{p})$, the minimum realization of $\bar{v}_{2}$ is
$\vee_{2}\in V^{r}({\mathbb{R}}^{q})$. Then the minimum realization of
$\bar{v}_{1}$ is $v_{1}\in V^{r}({\mathbb{R}}^{p\vee q})$. Then the integral
curve of $\bar{v}$ can be lifted as shown in Fig 3.

$t_{0}$$t_{1}$$t_{2}$$t$$\Omega$${\mathbb{R}}^{p}$${\mathbb{R}}^{q}$${\mathbb{R}}^{p\vee
q}$$\bar{v}_{0}$$v_{0}$$v_{0}$$\bar{v}_{1}$$v_{1}$$v_{1}$$v_{1}$$\bar{v}_{2}$$v_{2}$$v_{2}$
Figure 3: Lift and projection of integral curves

  * 1.

Docking:

Assume there are two original control systems as

$\displaystyle\Sigma_{1}:\quad\dot{x}=F(x,u),\quad
x\in{\mathbb{R}}^{m},\;u\in{\mathbb{R}}^{p};$ (164)
$\displaystyle\Sigma_{2}:\quad\dot{z}=G(z,v),\quad
z\in{\mathbb{R}}^{n},\;u\in{\mathbb{R}}^{q}.$ (165)

They will be docked into the target system $\Omega$ as

$\displaystyle\dot{\xi}=H(\xi,w),\quad\xi\in{\mathbb{R}}^{s},\;w\in{\mathbb{R}}^{\ell}.$
(166)

It is required that the docking happens during a transient period
$[T_{0},T_{1}]$, and the process is smooth.

###### Definition 7.115

System (164) and system (165) are said to be docked into (166) smoothly during
the transient period $[T_{0},T_{1}]$. If the following requirements are
satisfied.

  * (i)

There exists a smooth monotonically non-decreasing function $\lambda(t)$,
$t\in[T_{0},T_{1}]$, such that

$\lambda(t)=\begin{cases}0,\quad t=T_{0},\\\ 1,\quad t=T_{1}.\end{cases}$

  * (ii)

There exists a control deformation function $w=\varphi(u,v)$, and using it a
control system over $\Omega$, called a transient system, is constructed as

$\displaystyle\dot{\bar{\xi}}=(1-\lambda(t))\begin{bmatrix}\bar{F}(\bar{x},u)\\\
\bar{G}(\bar{z},v)\end{bmatrix}+\lambda(t)\bar{H}(\bar{\xi},w)+\Psi(\bar{x},\bar{z},\bar{\xi},u,v),\quad
t\in[T_{0},T_{1}].$ (167)

  * (iii)

The transient system is controllable to

$\begin{bmatrix}\bar{x}(T_{1})\\\
\bar{z}(T_{1})\end{bmatrix}=\bar{\xi}(T_{1}),$

with $\dim(\bar{\xi(T_{1})})=p\wedge q\wedge s$.

###### Remark 7.116

In (167) $\Psi(\bar{x},\bar{z},\bar{\xi},u,v)$ is a (fictitious) virtual
force, caused by the interaction of docking objects, satisfying

$\Psi(\bar{x},\bar{z},\bar{\xi},u,v)=0,\quad if\begin{bmatrix}\bar{x}\\\
\bar{z}\end{bmatrix}\vec{\vdash}\bar{\xi}=0.$

In control of Clutch System the virtual force is chosen as [23, 24]

$\tau_{c}=FcR_{a}\psi(\omega_{i},\omega_{0}).$

Particularly, it was chosen as [9]: $\tau_{c}=sign(\omega_{i}-\omega_{0})F$.

  * 1.

Undocking:

Assume there is an original system $\Sigma_{1}$ as:

$\displaystyle\dot{\xi}=H(\xi,w),\quad\xi\in{\mathbb{R}}^{s},\;w\in{\mathbb{R}}^{\ell}.$
(168)

It will be undocked into two systems as

$\displaystyle\Sigma_{2}:\quad\dot{x}=F(x,u),\quad
x\in{\mathbb{R}}^{m},\;u\in{\mathbb{R}}^{p};$ (169)
$\displaystyle\Sigma_{3}:\quad\dot{z}=G(z,v),\quad
z\in{\mathbb{R}}^{n},\;u\in{\mathbb{R}}^{q}.$ (170)

It is required that the docking happens during a transient period
$[T_{0},T_{1}]$, and the process is smooth.

###### Definition 7.117

System (168) is said to be un-docked into (169) and (170) smoothly during the
transient period $[T_{0},T_{1}]$. If the follow requirements are satisfied.

  * (i)

There exists a smooth monotonically non-decreasing function $\lambda(t)$,
$t\in[T_{0},T_{1}]$, such that

$\lambda(t)=\begin{cases}0,\quad t=T_{0},\\\ 1,\quad t=T_{1}.\end{cases}$

  * (ii)

There exist two control deformation functions $u=\varphi(w),~{}v=\varphi(w)$,
and using them a control system over $\Omega$, called a transient system, is
constructed as

$\displaystyle\begin{array}[]{rcl}\dot{\bar{\xi}}&=&(1-\lambda(t))\bar{H}(\bar{\xi},w)\\\
{}\hfil&{}\hfil&+\lambda(t)\begin{bmatrix}\bar{F}(\bar{x},u)\\\
\bar{G}(\bar{(}z),v)\end{bmatrix}+\Psi(\bar{x},\bar{z},\bar{\xi},w),~{}t\in[T_{0},T_{1}].\end{array}$
(173)

  * (iii)

The transient system is controllable to
$\bar{\xi}(T_{1})=\begin{bmatrix}\bar{x}(T_{1})\\\
\bar{z}(T_{1})\end{bmatrix}$, with $\dim(\bar{\xi(T_{1})})=p\wedge q\wedge s$.

###### Remark 7.118

In (173) $F(\bar{x},\bar{z},\bar{\xi},w)$ is a virtual force, caused by the
interaction of docking objects, satisfying

$\Psi(\bar{x},\bar{z},\bar{\xi},w)=0,\quad if\begin{bmatrix}\bar{x}\\\
\bar{z}\end{bmatrix}\vec{\vdash}\bar{\xi}=0.$

##  8 Concluding Remarks

The main purpose of this paper is to construct a new geometric object called
the DFES (or DFM), which provides a framework (i.e., the state space) for
DVDS. We briefly summarize the results.

The DFES is constructed as follows:

  * 1.

Step 1: Define an inner product for two vectors of different dimensions. It
turns the ESDD
${\mathcal{V}}={\mathbb{R}}^{\infty}=\bigcup_{n=1}^{\infty}{\mathbb{R}}^{n}$
into a distance space.

  * 2.

Step 2: Two vectors $x,y\in{\mathbb{R}}^{\infty}$ are said to be equivalent,
denoted by $x\leftrightarrow y$, if their distance is zero. The quotient space
$\Omega={\mathbb{R}}^{\infty}/\leftrightarrow$ is called the DFES. In fact,
$d_{{\mathcal{V}}}(x,y)=0$, if and only if, there exist ${\bf 1}_{\alpha}$ and
${\bf 1}_{\beta}$ such that $x\otimes{\bf 1}_{\alpha}=y\otimes{\bf
1}_{\beta}$. It is clear that two vectors are equivalent if they contain the
same information. In other words, vector form is a way for a set of data to
express itself. It may be expressed as vectors of different dimensions, but
from information point of view, they are equivalent.

  * 3.

Step 3: By posing scalar multiplication and addition $\Omega$ becomes a
topological real vector space. Let
$\Pr:{\mathbb{R}}^{\infty}\rightarrow\Omega$ be the natural projection. Then
$({\mathbb{R}}^{\infty},\Pr,\Omega)$ becomes a fiber bundle.

  * 4.

Step 4. Using the fiber bundle structure of
$({\mathbb{R}}^{\infty},\Pr,\Omega)$, each $\bar{x}\in\Omega$ has a coordinate
neighborhood, which is a set of coordinate charts of various dimensions. It is
called a bundle of coordinate charts. Hence $\Omega$ is called a DFES.

  * 5.

Step 5. Using the bundles of coordinate charts, a differentiable structure can
be posed on $\Omega$, making it a DFM. Then the continuous functions,
(co)-vector fields, (co)-distributions, and tensor fields can be built for
$\Omega$. Eventually, the dimension-free Riemannian manifolds and dimension-
free symplectic manifolds can be properly constructed.

Note that the gluing topology on the DFES
$\Omega={\mathbb{R}}^{\infty}/\leftrightarrow$ makes it a path-wise connected
topological space. Therefore, intuitively, the trajectories of dynamic systems
over $\Omega$ can continuously move “across” Euclidian spaces of different
dimensions. This is the main idea for using DFM to design DVDS and DVCS.
Lifting the trajectory of a dynamic system over $\Omega$ to a leaf (a
Euclidian space of fixed dimension) is called a realization. As a dynamic
system over $\Omega$ is lifted onto leafs of different dimensions, a
dimension-varying realization is obtained. Conversely, we can also project the
trajectory of a dynamic system on a Euclidian space of fixed dimension onto
$\Omega$.

The design of dimension-varying dynamic (control) systems can be described as
follows:

  * 1.

Step 1: Project a dimension-varying dynamic system, which has broken vector
fields over Euclidian spaces of different dimensions, onto $\Omega$ to form a
dynamic system over $\Omega$, which consists of several (finite number) of
vector fields.

  * 2.

Step 2: Lifting the dynamic system on $\Omega$ to a Euclidian space of proper
dimension, where all the vector fields involved by the dynamic system on
$\Omega$ can be properly lifted into this Euclidian space.

  * 3.

Step 3: All the analysis and control design can be done in conventional way
for this lifting system on its Euclidian space.

  * 4.

Step 4: Project the resulting manipulated system back to $\Omega$ and then
lifting it into several original Euclidean spaces, where the original
dimension-varying system lies on.

Finally, we would like to present a conjecture: The DFM might provide a
framework (i.e., the state space) for string theory in physics. The idea is
sketched as follows:

Consider a subspace of DFM as
$\Omega_{3}:=\left\\{\bar{x}\in\Omega\;|\;\dim(\bar{x})\leq 3\right\\}$. We
choose $3$ because it is the dimension of real physical world.

Now $({\mathbb{R}}^{\infty},Pr,\Omega_{3})$ is a sub-bundle of the fiber-
bundle $({\mathbb{R}}^{\infty},\Pr,\Omega)$. If we consider all possible
realization of dynamic systems over $\Omega_{3}$, then the minimum total
subspace which allow all possible realizations is ${\mathbb{R}}^{[\cdot,6]}$.
Hence, if we desire a space which is of minimum dimension and contains all
moves (or dynamic systems), then it is
${\mathcal{B}}:=({\mathbb{R}}^{[\cdot,6]}\rightarrow\Omega_{3})$. Since this
manifold is of dimension $9$, plus a dimension for $t$, a manifold of
dimension $10$ is reasonable for describing state-motion-time. This might be
the string space.

Some further arguments are the following:

  * (i)

It is well known in classical differential geometry that an $n$-dimension
manifold $M$ has an $n$-dimensional tangent space at each point. Hence, if
taking both $M$ and $T(M)$ into consideration, an $n$-dimensional manifold
with its tangent bundle is a $2n$ dimensional manifold, which is a well known
fact. So consider the bundle ${\mathcal{B}}$ as a $9$-dimensional manifold is
reasonable.

  * (ii)

It seems that there is no static particle in the world. That is, particles are
always joined with their moves. Moves can be described by vector fields. So to
describe a particle, a position plus a vector field on its tangent space may
be reasonable to describe it, as the particle is small and its movement is
very fast. Using string to describe a particle might essentially be a
description for both the position and the trajectory of a particle.

  * (iii)

Now taking the position and moving trajectory into consideration. We may
consider the extra $6$ dimensions being used to describe open string (or open
movement of a particle). In addition, we need $SU(3)$ to describe the gauge
group and $SU(1)$ for rotation. Then we have
${\mathcal{B}}+SU(3)+SU(1)+\mbox{time}$, which is of dimension $26$. This
manifold might be proper for Bosonic super-sting model.

In brief, DFM could provide a framework for systems with arbitrary dimensions.
DFM with a Reimannian structure becomes a DFRM. The investigation of DFRM in
this paper is very elementary. A continuous study is necessary. It is
promising that DFRM might overcome the crisis of classical Riemannian geometry
[29].

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