# Average orders of Goldbach Estimates in Arithmetic Progressions

Thi Thu Nguyen  Université de Lille, Laboratoire Paul Painlevé, 59655
Villeneuve d’Ascq Cedex, France<EMAIL_ADDRESS>thithu.nguyen@univ-
lille.fr

###### Abstract.

We obtain asymptotic results on the average numbers of Goldbach
representations of an interger as the sum of two primes in different
arithmetic progressions. We also prove an omega-result showing that the
asymptotic result is essentially the best possible.

###### Key words and phrases:

Goldbach numbers, congruences, Dirichlet L-function, Generalized Riemann
Hypothesis, Cesàro weight

###### 2020 Mathematics Subject Classification:

Primary 11P32; Secondary 11P55

##  1\. Introduction

The Goldbach conjecture that every even integer greater than two can be
written as the sum of two primes, is one of the oldest open problems today.
Instead of studying directly the Goldbach counting function, we consider the
corresponding problem for a smoother version using logarithms, that is the
weighted Goldbach function

$G(n)=\sum_{m+l=n}\Lambda(m)\Lambda(l),$

where $\Lambda(n)$ is the von Mangoldt function. We can get a lot of
information about $G(n)$ by studying its average order, that is

$S(X):=\sum_{n\leq X}G(n).$

It was first studied by Fujii [7] in the following theorem

######  Theorem 1.1.

Suppose that the Riemann Hypothesis (RH) is true. Then we have

$S(X)=\dfrac{X^{2}}{2}+H(X)+\tilde{E}(X),$

where $H(X)=-2\sum_{\rho}\dfrac{x^{\rho+1}}{\rho(\rho+1)}$ is the sum over the
non-trivial zeros $\rho$ of the Riemann zeta function, counted with
multiplicity and $\tilde{E}(X)\ll(X\log X)^{4/3}$.

######  Remark 1.2.

Assuming the RH, the order of magnitude of $H(X)$ is $\mathcal{O}(X^{3/2})$,
while unconditionally, it is

$H(X)\ll X^{1+B},$

where $B=\sup\\{Re(\rho):\zeta(\rho)=0\\}.$

Granville recovered Fujii’s result without RH, $\tilde{E}(X)\ll
X^{\frac{2+4B}{3}}\log^{2}X$ [11, 12]. Egami and Matsumoto expected the error
term to be $\mathcal{O}(X^{1+\varepsilon})$ ([6, Conjecture 2.2]). In fact
this conjecture was proved by Bhowmik and Schlage-Puchta [2] who showed in
2010 that the error term is $\mathcal{O}(X\log^{5}X)$ under RH and
$\Omega(X\log\log X)$ unconditionally. It was improved by Languasco-Zaccagnini
[15] in 2012 and using different methods later by Goldson-Yang [10] in 2017 or
Goldston-Suriajaya [9] in 2023 to $\tilde{E}(X)\ll X\log^{3}X.$

This paper studies the case of two primes in arithmetic progressions which was
first introduced by Rüppel [19] in 2012. Suzuki [21] in 2017 obtained an
asymptotic formula under the Generalized Riemann Hypothesis with real zeros
while for the case of common modulus Bhowmik-Halupczok-Matsumoto-Suzuki [1]
showed an asymptotic formula independent of Generalized Riemann Hypothesis
(GRH).

Let $q_{1},q_{2}$ be positive integers and $1\leq a_{1}<q_{1},1\leq
a_{2}<q_{2}$ be positive integers such that $(a_{1},q_{1})=1$,
$(a_{2},q_{2})=1$. We consider the function

$G(n,q_{1},q_{2},a_{1},a_{2}):=\sum_{\begin{subarray}{c}m+l=n\\\ m\equiv
a_{1}(q_{1})\\\ l\equiv a_{2}(q_{2})\end{subarray}}\Lambda(m)\Lambda(l),$

whose summatory function is defined as

$S(X,q_{1},q_{2},a_{1},a_{2})=\sum_{n\leq X}G(n,q_{1},q_{2},a_{1},a_{2}).$

We denote by $\chi$ a Dirichlet character modulo $q$. Let $\rho_{\chi}$ be the
non-trivial zeros of $L(s,\chi)$ and let
$B_{\chi}=\sup\\{\Re_{\rho_{\chi}}\\}$, $B_{q}=\sup\\{B_{\chi}\mid\chi(q)\\}$.

In this article, we prove an asymptotic formula for
$S(X,q_{1},q_{2},a_{1},a_{2})$ and an omega-result, which can be stated as
follows.

######  Theorem 1.3.

  1. (A)

For any $\varepsilon>0$

(1)
$\displaystyle\begin{split}S(X,q_{1},q_{2},a_{1},a_{2})&=\dfrac{X^{2}}{2\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}H(X,q_{1},a_{1})\\\
&+\dfrac{1}{\varphi(q_{1})}H(X,q_{2},a_{2})+\mathcal{O}(X^{B^{*}_{q_{1}}+B^{*}_{q_{2}}}\log
X\log(q_{1}X)\log(q_{2}X)),\end{split}$

where the implicit constant depends on $q_{1},q_{2}$ and

$H(X,q,a)=-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\rho_{\chi}}\dfrac{X^{{\rho_{\chi}}+1}}{{\rho_{\chi}}({\rho_{\chi}}+1)},$
$B^{*}_{q}=B^{*}_{q}(X)=\min(B_{q},1-\eta),\hskip
28.45274pt\eta=\eta_{q}(X)=\dfrac{c_{1}(\varepsilon)}{q^{\varepsilon}+(\log
X)^{2/3}(\log\log X)^{1/3}},$

with some small constant $c_{1}(\varepsilon)>0$.

  2. (B)

We have

(2)
$\displaystyle\begin{split}S(X,q_{1},q_{2},a_{1},a_{2})&=\dfrac{X^{2}}{2\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}H(X,q_{1},a_{1})\\\
&+\dfrac{1}{\varphi(q_{1})}H(X,q_{2},a_{2})+\Omega(X\log\log X),\end{split}$

where the implicit constant depends on $q_{1},q_{2}$.

######  Remark 1.4.

For the unconditional order of magnitude of $H(x,q,a)$, we have

$\displaystyle H(X,q,a)$
$\displaystyle=-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\rho_{\chi}}\dfrac{X^{{\rho_{\chi}}+1}}{{\rho_{\chi}}({\rho_{\chi}}+1)}$
$\displaystyle\ll\dfrac{X^{1+B_{q}}}{\varphi(q)}\sum_{\chi(q)}\sum_{\rho_{\chi}}\dfrac{1}{|{\rho_{\chi}}({\rho_{\chi}}+1)|}.$

By using [1, Lemma 3], we obtain

$\displaystyle H(X,q,a)$
$\displaystyle\ll\dfrac{X^{1+B_{q}}}{\varphi(q)}\sum_{\chi(q)}\left(\log^{2}q+\delta_{1}(\chi)q^{1/2}\log^{2}q\right)$
$\displaystyle\ll X^{1+B_{q}}\log^{2}q,$

where $\delta_{1}(\chi)=1$ if $\chi$ is the exceptional character and
$\delta_{1}(\chi)=0$ otherwise. Assuming GRH, we can obtain

$H(X,q,a)\ll_{q}X^{3/2}.$

######  Corollary 1.5.

Assume the GRH. Then we have

(3)
$\displaystyle\begin{split}S(X,q_{1},q_{2},a_{1},a_{2})&=\dfrac{X^{2}}{2\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}H(X,q_{1},a_{1})+\dfrac{1}{\varphi(q_{1})}H(X,q_{2},a_{2})\\\
&+\mathcal{O}(X\log X\log(q_{1}X)\log(q_{2}X)),\end{split}$

where the implicit constant depends on $q_{1},q_{2}$.

######  Remark 1.6.

Assume GRH, we can see that Corollary 1.5 is similar to [21, Theorem 1.1] .

When $q_{1}=q_{2}=q$, we replace $S(X,q_{1},q_{2},a_{1},a_{2})$ by
$S(X,q,a_{1},a_{2})$. Then we obtain the following corollary.

######  Corollary 1.7.

Let $a_{1},a_{2}$ be integers with $(a_{1}a_{2},q)=1$, we have

$\displaystyle S(X,q,a_{1},a_{2})$
$\displaystyle=\dfrac{X^{2}}{2\varphi^{2}(q)}-\dfrac{1}{\varphi^{2}(q)}\sum_{\chi(q)}(\overline{\chi}(a_{1})+\overline{\chi}(a_{2}))\sum_{\rho_{\chi}}\dfrac{X^{{\rho_{\chi}}+1}}{{\rho_{\chi}}({\rho_{\chi}}+1)}$
$\displaystyle+\mathcal{O}(X^{2B^{*}_{q}}\log X\log(qX)^{2}),$

where the implicit constant may depend on $q$.

This result improves the upper bound $\log^{5}qX$ of [1, Theorem 2] by a
factor $\frac{\log(qX)^{3}}{\log X}$. When we take $q=1$, we recover the best
known result for $S(X)$.

Now, we put $z=re(\alpha)$, $r=e^{-1/X}$, $|\alpha|\leq 1/2$. Let $q$ be a
positive integer, $1\leq a<q$ be an integer coprime to $q$ and $1\leq y\leq
X$, let us consider a power series as was done by Hardy-Littlewood [13].

$\displaystyle F_{a,q}(z):=\sum_{\begin{subarray}{c}n\geq 1\\\ n\equiv
a(q)\end{subarray}}\Lambda(n)z^{n}$ $\displaystyle
I_{q}(z):=\dfrac{1}{\varphi(q)}\sum_{n\geq 1}z^{n}\hskip
28.45274ptI(y,z):=\sum_{n\leq y}z^{n}.$

Our choice is natural because $F_{a,q}(z)$ is a generating function of primes
in arithmetic progression with additive property and we expect it to be close
to $I_{q}(z)$ by Dirichlet’s theorem.

We will show that $S(X,q_{1},q_{2},a_{1},a_{2})$ can be expressed in terms of
$F_{a_{1},q_{1}}(z)$, $F_{a_{2},q_{2}}(z)$ in the following theorem.

######  Theorem 1.8.

We have

$\displaystyle S(X,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle=\dfrac{X^{2}}{2\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}H(X,q_{1},a_{1})+\dfrac{1}{\varphi(q_{1})}H(X,q_{2},a_{2})$
$\displaystyle+\mathcal{O}(X(\log^{2}(q_{1}X)+\log^{2}(q_{2}X)))+E(X,q_{1},q_{2},a_{1},a_{2}),$

where

(4) $\displaystyle
E(X,q_{1},q_{2},a_{1},a_{2})=\int^{1}_{0}\left(F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right)\left(F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right)I\left(X,\dfrac{1}{z}\right)d\alpha,$

and the implicit constant depends on $q_{1},q_{2}$.

Next, Languasco-Zaccagnini [16] introduced a Cesàro weighted version for
Goldbach numbers. That is, for $X\geq 2$, and $k>1$ a real number

$\displaystyle\sum_{n\leq
X}G(n)\dfrac{\left(1-\frac{n}{X}\right)^{k}}{\Gamma(k+1)}$
$\displaystyle=\dfrac{X^{2}}{\Gamma(k+3)}-2\sum_{\rho}\dfrac{\Gamma(\rho)}{\Gamma(\rho+k+2)}X^{\rho+1}$
$\displaystyle+\sum_{\rho_{1}}\sum_{\rho_{2}}\dfrac{\Gamma(\rho_{1})\Gamma(\rho_{2})}{\Gamma(\rho_{1}+\rho_{2}+k+1)}X^{\rho_{1}+\rho_{2}}+\mathcal{O}_{k}(X^{1/2}).$

In the study of these problems, the parameter $k$ plays a crucial role and we
would like to keep it as small as possible. The work of Languasco-Zaccagnini
treats $k>1$. Goldston-Yang [10] extended the above for $k=1$ under Riemann
Hypothesis. However, Cantarini proved this problem for $k=1$ without any
supported hypothesis [4]. After that, Brüdern-Kaczorowski-Perelli [3] gave a
general formula for $k>0$, still for $q=1$, in which the zero-sum converges
conditionally.

In arithmetic progressions, Cantarini-Gambini-Zaccagnini [5] studied the
Cesàro weighted version for $q_{1}=q_{2}$ and $k>1$. We would like to extend
this problem with $q_{1},q_{2}$ different and the most optimal $k$. So we
obtain the case $k=1$ in the following theorem.

######  Theorem 1.9.

When $X$ tends to infinity, we have

$\displaystyle\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)G(n,q_{1},q_{2},a_{1},a_{2})=\dfrac{X^{2}}{6\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}L(X,q_{1},a_{1})$
$\displaystyle+\dfrac{1}{\varphi(q_{1})}L(X,q_{2},a_{2})+\mathcal{O}(X^{B^{*}_{q_{1}}+B^{*}_{q_{2}}}\log
X\log(q_{1}X)\log(q_{2}X)),$

where the implicit constant depends on $q_{1},q_{2}$ and for $(a,q)=1$,

$\displaystyle
L(X,q,a)=-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\rho_{\chi}}\dfrac{X^{{\rho_{\chi}}+1}}{{\rho_{\chi}}({\rho_{\chi}}+1)({\rho_{\chi}}+2)}.$

######  Remark 1.10.

When $q_{1}=q_{2}=1$ and assuming RH, the error term in Theorem 1.9 is
$\mathcal{O}(X\log^{3}X)$. It is weaker than the result of Goldston-Yang [10]
because some functions which we used in this paper, are estimated better when
$q=1$ and under RH.

To prove the part (A) Theorem 1.3, we use the circle method to compute the
sum. Firstly, we will prove Theorem 1.8.

##  2\. Proof of Theorem 1.8

In this part we consider the generating function

$F_{a,q}(z):=\sum_{\begin{subarray}{c}n\geq 1\\\ n\equiv
a(q)\end{subarray}}\Lambda(n)z^{n}=\sum_{\begin{subarray}{c}n\geq 1\\\ n\equiv
a(q)\end{subarray}}\Lambda(n)e^{-n/X}e(n\alpha).$

We first note that

(5)
$\displaystyle\begin{split}\left(F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right)\left(F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right)&=F_{a_{1},q_{1}}(z)F_{a_{2},q_{2}}(z)-F_{a_{1},q_{1}}(z)I_{q_{2}}(z)\\\
&-F_{a_{2},q_{2}}(z)I_{q_{1}}(z)+I_{q_{1}}(z)I_{q_{2}}(z).\end{split}$

From the definitions of $I_{q}(z)$ and $F_{a,q}(z)$, we calculate

$\displaystyle
F_{a_{1},q_{1}}(z)F_{a_{2},q_{2}}(z)=\sum_{\begin{subarray}{c}m\geq 1\\\
m\equiv
a_{1}(q_{1})\end{subarray}}\Lambda(m)z^{m}\sum_{\begin{subarray}{c}k\geq 1\\\
k\equiv a_{2}(q_{2})\end{subarray}}\Lambda(k)z^{k}=\sum_{n\geq
2}G(n,q_{1},q_{2},a_{1},a_{2})z^{n}$ $\displaystyle
F_{a_{1},q_{1}}(z)I_{q_{2}}(z)=\dfrac{1}{\varphi(q_{2})}\sum_{\begin{subarray}{c}m,k\geq
1\\\ m\equiv
a_{1}(q_{1})\end{subarray}}\Lambda(m)z^{m+k}=\dfrac{1}{\varphi(q_{2})}\sum_{n\geq
2}\psi(n-1,q_{1},a_{1})z^{n},$ $\displaystyle
F_{a_{2},q_{2}}(z)I_{q_{1}}(z)=\dfrac{1}{\varphi(q_{1})}\sum_{n\geq
2}\psi(n-1,q_{2},a_{2})z^{n}$ $\displaystyle
I_{q_{1}}(z)I_{q_{2}}(z)=\dfrac{1}{\varphi(q_{1})\varphi(q_{2})}\sum_{n\geq
2}(n-1)z^{n},$

where $\psi$ is the Chebyshev function in arithmetic progression, defined as

$\psi(t,q,a)=\sum_{\begin{subarray}{c}m\leq t\\\ m\equiv
a(q)\end{subarray}}\Lambda(m).$

Hence, the right hand side of (5) can be rewritten as

$\sum_{n}B(n,q_{1},q_{2},a_{1},a_{2})r^{n}e(n\alpha),$

with

(6)
$\displaystyle\begin{split}B(n,q_{1},q_{2},a_{1},a_{2})&=G(n,q_{1},q_{2},a_{1},a_{2})-\dfrac{1}{\varphi(q_{2})}\sum_{n\geq
2}\psi(n-1,q_{1},a_{1})\\\ &-\dfrac{1}{\varphi(q_{1})}\sum_{n\geq
2}\psi(n-1,q_{2},a_{2})+\dfrac{1}{\varphi(q_{1})\varphi(q_{2})}\sum_{n\geq
2}(n-1)z^{n}.\end{split}$

So that

$\displaystyle E(X,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle=\int_{0}^{1}\sum_{n}\sum_{m\leq
X}B(n,q_{1},q_{2},a_{1},a_{2})r^{n}e(n\alpha)r^{-m}e(-m\alpha)d\alpha$
$\displaystyle=\sum_{n}\sum_{m\leq
X}B(n,q_{1},q_{2},a_{1},a_{2})r^{n-m}\int_{0}^{1}e((n-m)\alpha)d\alpha$
$\displaystyle=\sum_{n\leq X}B(n,q_{1},q_{2},a_{1},a_{2})$

since

$\int_{0}^{1}e(t\alpha)d\alpha=1\text{ if $t=0$ and }0\text{ otherwise.}$

Then we obtain the equation

(7)
$\displaystyle\begin{split}S(X,q_{1},q_{2},a_{1},a_{2})&=E(X,q_{1},q_{2},a_{1},a_{2})+\dfrac{1}{\varphi(q_{2})}\sum_{n\leq
X}\psi(n-1,q_{1},a_{1})\\\ &+\dfrac{1}{\varphi(q_{1})}\sum_{n\leq
X}\psi(n-1,q_{2},a_{2})-\sum_{n\leq
X}\dfrac{n-1}{\varphi(q_{1})\varphi(q_{2})}.\end{split}$

Moreover, we have an explicit formula for the average of $\psi(n,q,a)$ as
follows.

######  Lemma 2.1.

Let $X\geq 2$ and $(a,q)=1$, we have

$\displaystyle\sum_{n\leq
X}\psi(n-1,q,a)=\dfrac{X^{2}}{2\varphi(q)}-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\rho_{\chi}}\dfrac{X^{\rho_{\chi}+1}}{\rho_{\chi}(\rho_{\chi}+1)}+\mathcal{O}(X\log^{2}(qX)).$

###### Proof.

We first remark that

(8) $\displaystyle\sum_{n\leq X}\psi(n-1,q,a)=\sum_{\begin{subarray}{c}n\leq
X\\\ n\equiv a(q)\end{subarray}}(X-n)\Lambda(n)=\int_{1}^{X}\psi(t,q,a)dt.$

Now let us recall of [14, (5.66)],

$\displaystyle\psi(t,q,a)=\dfrac{t}{\varphi(q)}-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{{\mid\gamma_{\chi}\mid\leq
T}}\dfrac{t^{\rho_{\chi}}-1}{\rho_{\chi}}+\mathcal{O}(\dfrac{t}{T}\log^{2}(qt)),$

for $q\geq 1$, $(a,q)=1$ and $1\leq T\leq t$ with an absolute implicit
constant.

By using [1, Lemma 2],

$\displaystyle\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{1}{|\rho_{\chi}|}$
$\displaystyle\ll\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\left(\log(2qT)^{2}+\delta_{1}(\chi)q^{1/2}\log^{2}q\right)$
$\displaystyle\ll\log(2qT)^{2},$

since there is at most one exceptional character mod $q$.

Hence, we can rewrite

(9)
$\displaystyle\psi(t,q,a)=\dfrac{t}{\varphi(q)}-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}+\mathcal{O}(\dfrac{t}{T}\log^{2}(qt)).$

Substituting (9) into (8),

$\displaystyle\sum_{n\leq X}\psi(n-1,q,a)$
$\displaystyle=\dfrac{X^{2}}{2\varphi(q)}-\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{X^{\rho_{\chi}+1}}{\rho_{\chi}(\rho_{\chi}+1)}$
$\displaystyle+\mathcal{O}(\dfrac{X^{2}}{T}\log^{2}(qX)).$

We remark that

$\displaystyle\sum_{\mid\gamma_{\chi}\mid\geq
T}\dfrac{X^{\rho_{\chi}+1}}{\rho_{\chi}(\rho_{\chi}+1)}\ll
X^{2}\sum_{\mid\gamma_{\chi}\mid\geq
T}\dfrac{1}{|\rho_{\chi}|^{2}}\ll\dfrac{X^{2}}{T}\log(qT).$

Therefore Lemma 2.1 is proved when we choose $T=X$. ∎

The proof of Theorem 1.8 is complete when we substitute Lemma 2.1 into (7).

##  3\. Proof of the part (A) Theorem 1.3

From Theorem 1.8, to prove (A) Theorem 1.3, it is enough to estimate

$E(X,q_{1},q_{2},a_{1},a_{2})\ll_{q_{1},q_{2}}X^{B^{*}_{q_{1}}+B^{*}_{q_{2}}}\log
X\log(q_{1}X)\log(q_{2}X).$

By the Cauchy-Schwarz inequality, the integral $E(X,q_{1},q_{2},a_{1},a_{2})$
is estimated as

$\displaystyle|E(X,q_{1},q_{2},a_{1},a_{2})|^{2}=\left|\int^{1}_{0}\left(F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right)\left(F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right)I\left(X,\dfrac{1}{z}\right)d\alpha\right|^{2}$
$\displaystyle\ll\int^{1}_{0}\left|F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right|^{2}\left|I\left(X,\dfrac{1}{z}\right)\right|d\alpha\int^{1}_{0}\left|F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right|^{2}\left|I\left(X,\dfrac{1}{z}\right)\right|d\alpha.$

We remark that $|E(X,q_{1},q_{2},a_{1},a_{2})|$ can be estimated as the
product of two factors which have the same properties.

So we need only consider

(10) $\displaystyle
T_{1}(X,q,a):=\int^{1}_{0}\left|F_{a,q}(z)-I_{q}(z)\right|^{2}\left|I\left(X,\dfrac{1}{z}\right)\right|d\alpha,$

we remark that

$|E(X,q_{1},q_{2},a_{1},a_{2}))|\ll\max_{q_{i},a_{i}}(T_{1}(X,q,a)).$

We will estimate $T_{1}(X,q,a)$ by bounding $I\left(X,\dfrac{1}{z}\right)$ and
then the moment of $\left|F_{a,q}(z)-I(z)\right|$. We first remark that , for
$1\leq y\leq X$, by partial summation,

$I\left(y,\dfrac{1}{z}\right)=\sum_{n\leq
y}e^{n/X}e(-n\alpha)=e^{y/X}\sum_{n\leq
y}e(-n\alpha)-\dfrac{1}{X}\int_{1}^{y}\sum_{n\leq t}e(-n\alpha)e^{t/X}dt.$

Since

$\sum_{n\leq
t}e(-n\alpha)\ll\min\left(t,\dfrac{1}{\left|\alpha\right|}\right),$

we obtain

(11) $\displaystyle
I\left(y,\dfrac{1}{z}\right)\ll\min\left(y,\dfrac{1}{\left|\alpha\right|}\right).$

We now prove the following lemma.

######  Lemma 3.1.

Let $X\geq 2$ and $\left|\alpha\right|\leq 1/2$. For any integrable positive
function of period 1, we have

(12)
$\displaystyle\int^{1/2}_{-1/2}f(\alpha)\min\left(X,\dfrac{1}{\left|\alpha\right|}\right)d\alpha\ll\sum_{k=0}^{\mathcal{O}(\log
X)}\frac{X}{2^{k}}\int_{0}^{2^{k+1}/X}f({\alpha})d\alpha.$

###### Proof.

Because of periodicity, we can restrict ourselves to $[0,1/2]$. We evaluate
the left hand side of (12)

$\displaystyle\int^{1/2}_{0}f(\alpha)\min\left(X,\dfrac{1}{\left|\alpha\right|}\right)d\alpha=X\int^{1/X}_{0}f(X)d\alpha+\int^{1/2}_{1/X}f(\alpha)\dfrac{d\alpha}{\alpha}.$

We write $[\frac{1}{X},\frac{1}{2}]$ as the disjoint union of
$[\frac{2^{k}}{X},\frac{2^{k+1}}{X}]$, for $0\leq k\leq\mathcal{O}(\log X)$.
Then we rewrite the right hand side of the above formula as

$\displaystyle X\int^{1/X}_{0}f(\alpha)d\alpha+\sum_{k=0}^{\mathcal{O}(\log
X)}\int^{2^{k+1}/X}_{2^{k}/X}f(\alpha)\dfrac{d\alpha}{\alpha}$
$\displaystyle\leq
X\int^{1/X}_{0}f(\alpha)d\alpha+\sum_{k=0}^{\mathcal{O}(\log
X)}\dfrac{X}{2^{k}}\int^{2^{k+1}/X}_{2^{k}/X}f(\alpha)d\alpha$
$\displaystyle\ll\sum_{k=0}^{\mathcal{O}(\log
X)}\dfrac{X}{2^{k}}\int^{2^{k+1}/X}_{0}f(\alpha)d\alpha.$

∎

Since $z=z(\alpha)=e^{-1/X}e(n\alpha)$, we replace
$f(\alpha)=\left|F_{a,q}(z)-I(z)\right|^{2}$ in Lemma 3.1, to obtain

(13) $\displaystyle T_{1}(X,q,a)$
$\displaystyle\ll\sum_{k=0}^{\mathcal{O}(\log
X)}\frac{X}{2^{k}}\int_{0}^{2^{k+1}/X}\left|F_{a,q}(z)-I(z)\right|^{2}d\alpha.$

For $1\leq h\leq X$, putting

$\displaystyle W(X,q,a,h):$
$\displaystyle=\int_{0}^{1/2h}\left|F_{a,q}(z)-I(z)\right|^{2}d\alpha$
$\displaystyle=\int_{0}^{1/2h}\left|\sum_{n}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}e(n\alpha)\right|^{2}d\alpha,$

where $\delta(n)$ is defined as

$\delta(n)=1\text{ if }n\equiv a(\text{mod }q)\text{ and }0\text{ otherwise}.$

Using [17, Gallagher’s lemma], for $1\leq h\leq X$, we obtain

(14)
$\displaystyle\begin{split}W(X,q,a,h)&\ll\dfrac{1}{h^{2}}\int_{-h}^{\infty}\left|\sum_{x<n\leq
x+h}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}\right|^{2}dx\\\
&\ll\dfrac{1}{h^{2}}\int_{0}^{h}\left|\sum_{n\leq
x}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}\right|^{2}dx\\\
&+\dfrac{1}{h^{2}}\int_{0}^{\infty}\left|\sum_{x<n\leq
x+h}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}\right|^{2}dx.\end{split}$

Next we evaluate

$\displaystyle J_{1,a,q}(X,h):=\int_{0}^{h}\left|\sum_{n\leq
x}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}\right|^{2}dx$
$\displaystyle J_{2,a,q}(X,h):=\int_{0}^{\infty}\left|\sum_{x<n\leq
x+h}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}\right|^{2}dx$

from the two following lemmas.

######  Lemma 3.2.

Let $X\geq 2$ and $1\leq h\leq X$,

$\displaystyle J_{1,a,q}(X,h)\ll H_{a,q}(h)+\dfrac{h}{\varphi^{2}(q)},$

where

$H_{a,q}(x)=\int_{0}^{x}\left(\psi(t,q,a)-\dfrac{t}{\varphi(q)}\right)^{2}dt.$

###### Proof.

We have

(15) $\displaystyle\begin{split}J_{1,a,q}(X,h)&=\int_{0}^{h}\left|\sum_{n\leq
x}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)e^{-n/X}\right|^{2}dx\\\
&\leq\int_{0}^{h}\left|\sum_{n\leq
x}\left(\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}\right)\right|^{2}dx=\int_{0}^{h}\left(\psi(x,q,a)-\dfrac{[x]}{\varphi(q)}\right)^{2}dx\\\
&\ll\int_{0}^{h}\left(\psi(x,q,a)-\dfrac{x}{\varphi(q)}\right)^{2}dx+\dfrac{1}{\varphi(q)^{2}}\int_{0}^{h}(x-[x])^{2}dx\\\
&\leq H_{a,q}(h)+\dfrac{h}{\varphi^{2}(q)}.\end{split}$

∎

######  Lemma 3.3.

Let $X\geq 2$ and $1\leq h\leq X$,

$\displaystyle
J_{2,a,q}(X,h)\ll\dfrac{h^{2}}{X^{2}}\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(H_{a,q}(jX)+\dfrac{jX}{\varphi^{2}(q)}\right)+\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(K_{a,q}(jX,h)+\dfrac{jX}{\varphi^{2}(q)}\right),$

where

$K_{a,q}(x,h)=\int_{0}^{x}\left(\psi(t+h,q,a)-\psi(t,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt$

and $H$ is as defined in the last lemma.

###### Proof.

We denote

$F(n):=\Lambda(n)\delta(n)-\dfrac{1}{\varphi(q)}.$

For $y\geq 2$, using partial summation, we have

(16) $\displaystyle\sum_{n\leq y}F(n)e^{-n/X}=e^{-y/X}\sum_{n\leq
y}F(n)+\dfrac{1}{y}\int_{0}^{y}\sum_{n\leq t}F(n)e^{-t/X}dt.$

We replace $y$ in (16) by $x+h$ and $x$, we obtain

(17) $\displaystyle\begin{split}&\sum_{x<n\leq
x+h}F(n)e^{-n/X}=\left(\sum_{n\leq x+h}-\sum_{n\leq x}\right)F(n)e^{-n/X}\\\
&=e^{-(x+h)/X}\sum_{n\leq x+h}F(n)+\dfrac{1}{X}\int_{0}^{x+h}\sum_{n\leq
t}F(n)e^{-t/X}dt\\\ \end{split}$ $\displaystyle-e^{-x/X}\sum_{n\leq
x}F(n)-\dfrac{1}{X}\int_{0}^{x}\sum_{n\leq t}F(n)e^{-t/X}dt$
$\displaystyle=e^{-x/X}\sum_{x<n\leq x+h}F(n)+e^{-x/X}\sum_{n\leq
x+h}F(n)(e^{-h/X}-1)$ $\displaystyle+\dfrac{1}{X}\int_{x}^{x+h}\sum_{n\leq
t}F(n)e^{-t/X}dt.$

Since $e^{-t}-1\leq t$ for all $t$ non negative, then $e^{-h/X}-1\leq h/X$.
Now squaring and integrating (17), we obtain

(18)
$\displaystyle\begin{split}J_{2,a,q}(X,h)&=\int_{0}^{\infty}\left|\sum_{x<n\leq
x+h}F(x)e^{-n/X}\right|^{2}dx\\\ &\ll\int_{0}^{\infty}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}e^{-2x/X}dx+\dfrac{h^{2}}{X^{2}}\int_{0}^{\infty}\left(\sum_{n\leq
x+h}F(n)\right)^{2}e^{-2x/X}dx\\\
&+\dfrac{1}{X^{2}}\int_{0}^{\infty}\left(\int_{x}^{x+h}\sum_{n\leq
t}F(n)e^{-t/X}dt\right)^{2}dx.\end{split}$

Step 1: The first term on the right hand side of (18) is

$\displaystyle\int_{0}^{\infty}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}e^{-2x/X}dx=\sum_{j=0}^{\infty}\int_{jX}^{(j+1)X}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}e^{-2x/X}dx$
$\displaystyle\leq\sum_{j=0}^{\infty}e^{-2j}\int_{jX}^{(j+1)X}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}dx\leq\sum_{j=0}^{\infty}\dfrac{1}{2^{j}}\int_{0}^{(j+1)X}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}dx,$

since $e^{-2j}\leq 2^{-j}$ for all $j$ non-negative integers. Moreover, for
$y\geq 2$,

$\displaystyle\int_{0}^{y}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}dx=\int_{0}^{y}\left(\psi(x+h,q,a)-\psi(x,q,a)-\dfrac{[x+h]}{\varphi(q)}+\dfrac{[x]}{\varphi(q)}\right)^{2}dt$
$\displaystyle\ll\int_{0}^{y}\left(\psi(x+h,q,a)-\psi(x,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt+\int_{0}^{y}\left(\dfrac{[x+h]}{\varphi(q)}-\dfrac{[x]}{\varphi(q)}-\dfrac{h}{\varphi(q)}\right)^{2}dt$
$\displaystyle\ll K_{a,q}(y,h)+\dfrac{y}{\varphi^{2}(q)}.$

Hence,

(19) $\displaystyle\int_{0}^{\infty}\left(\sum_{x<n\leq
x+h}F(n)\right)^{2}e^{-2x/X}dx\ll\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(K_{a,q}(jX,h)+\dfrac{jX}{\varphi^{2}(q)}\right).$

Step 2: Next we consider the last term of (18). By the Cauchy-Schwarz
inequality,

$\displaystyle\int_{0}^{\infty}\left(\int_{x}^{x+h}\sum_{n\leq
t}F(n)e^{-t/X}dt\right)^{2}dx$ $\displaystyle\ll
h\int_{0}^{\infty}\int_{x}^{x+h}\left(\sum_{n\leq
t}F(n)\right)^{2}e^{-2t/X}dtdx$
$\displaystyle=h\int_{0}^{\infty}\left(\sum_{n\leq
t}F(n)\right)^{2}e^{-2t/X}\left(\int_{t-h}^{t}dx\right)dt$
$\displaystyle=h^{2}\int_{0}^{\infty}\left(\sum_{n\leq
t}F(n)\right)^{2}e^{-2t/X}dt.$

Moreover, similar to Step 1, we have

$\displaystyle\int_{0}^{\infty}\left(\sum_{n\leq t}F(n)\right)^{2}e^{-2t/X}dt$
$\displaystyle\leq\sum_{j=0}^{\infty}\dfrac{1}{2^{j}}\int_{0}^{(j+1)X}\left(\sum_{n\leq
t}F(n)\right)^{2}dt$
$\displaystyle\ll\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(H_{a,q}(jX)+\dfrac{jX}{\varphi^{2}(q)}\right).$

Then the last term of (18) is

(20)
$\displaystyle\ll\dfrac{h^{2}}{X^{2}}\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(H_{a,q}(jX)+\dfrac{jX}{\varphi^{2}(q)}\right).$

Step 3: Finally, the second term of (18) is

(21) $\displaystyle\dfrac{h^{2}}{X^{2}}\int_{0}^{\infty}\left(\sum_{n\leq
x+h}F(n)\right)^{2}e^{-2x/X}dx=e^{2h/X}\dfrac{h^{2}}{X^{2}}\int_{0}^{\infty}\left(\sum_{n\leq
t}F(n)\right)^{2}e^{-2t/X}dt$ $\displaystyle\hskip
170.71652pt\ll\dfrac{h^{2}}{X^{2}}\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(H_{a,q}(jX)+\dfrac{jX}{\varphi^{2}(q)}\right).$

Using (19), (20), (21) and (17), the proof is complete. ∎

Hence, it suffices to estimate the two functions $H_{a,q}(x)$ and
$K_{a,q}(x,h)$.

######  Lemma 3.4.

With the notation of Theorem 1.3, we have the estimate

$H_{a,q}(x)\ll_{q}x^{2B^{*}_{q}+1}.$

###### Proof.

Using (9), we can rewrite

(22)
$\displaystyle\begin{split}H_{a,q}(x)&=\dfrac{1}{\varphi^{2}(q)}\int_{0}^{x}\left(\sum_{\chi(q)}\overline{\chi}(a)\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right)^{2}dt+\mathcal{O}\left(\int_{0}^{x}\dfrac{t^{2}}{T^{2}}\log^{4}(qt)dt\right)\\\
&\ll\dfrac{1}{\varphi^{2}(q)}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\int_{0}^{x}\left|\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt+\dfrac{x^{3}}{T^{2}}\log^{4}(qx).\end{split}$

We first consider

$\displaystyle\int_{0}^{x}\left|\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt\ll\int_{0}^{x}\left|\sum_{\mid\gamma_{\chi}\mid\leq
1}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt+\int_{0}^{x}\left|\sum_{1<\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt.$

By the well known bound of the Siegel zero, we have

$\displaystyle\int_{0}^{x}\left|\sum_{\mid\gamma_{\chi}\mid\leq
1}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt\ll_{q}x^{2B^{*}_{q}+1}.$

For the last sum, we get

$\displaystyle\int_{0}^{x}\left|\sum_{1<\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt$
$\displaystyle\ll\sum_{1<\mid\gamma_{\chi}\mid\leq
T}\sum_{1<\mid\gamma^{\prime}_{\chi}\mid\leq
T}\dfrac{x^{\beta_{\chi}+\beta^{\prime}_{\chi}+1}}{|\gamma_{\chi}||\gamma^{\prime}_{\chi}|(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)}$
$\displaystyle\ll x^{2B^{*}_{q}+1}\sum_{1<\mid\gamma_{\chi}\mid\leq
T}\sum_{1<\mid\gamma^{\prime}_{\chi}\mid\leq
T}\dfrac{1}{|\gamma_{\chi}||\gamma^{\prime}_{\chi}|(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)}.$

In view of the symmetry of zeros about the real axis, we may assume
$\gamma_{\chi}>0$. Now we consider the sum over $\gamma^{\prime}_{\chi}$, that

$\displaystyle\sum_{\gamma^{\prime}_{\chi}}\dfrac{1}{|\gamma^{\prime}_{\chi}|(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)}.$

We recall [17, Theorem 10.17] that for any $m\geq 0$,

(23) $\displaystyle\sum_{m\leq\mid\gamma_{\chi}\mid\leq m+1}1\ll\log(q(m+2)).$

Using (23), we will estimate the above sum for which
$\gamma^{\prime}_{\chi}<-\gamma_{\chi}$,
$|\gamma^{\prime}_{\chi}|\leq\frac{\gamma_{\chi}}{2}$,
$\frac{\gamma_{\chi}}{2}<\gamma^{\prime}_{\chi}<\frac{3\gamma_{\chi}}{2}$ and
$\gamma^{\prime}_{\chi}\geq\frac{3\gamma_{\chi}}{2}$ and obtain

(24)
$\displaystyle\sum_{\gamma^{\prime}_{\chi}}\dfrac{1}{|\gamma^{\prime}_{\chi}|(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)}\ll\dfrac{(\log
q\gamma_{\chi})^{2}}{\gamma_{\chi}}.$

Thus, we have

$\displaystyle\sum_{1<\gamma_{\chi}\leq
T}\sum_{\gamma^{\prime}_{\chi}}\dfrac{1}{|\gamma_{\chi}||\gamma^{\prime}_{\chi}|(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)}$
$\displaystyle\ll\sum_{1<\gamma_{\chi}\leq T}\dfrac{(\log
q\gamma_{\chi})^{2}}{\gamma_{\chi}^{2}}$
$\displaystyle\ll\sum_{n=1}^{T}\dfrac{\log^{3}(qn)}{n^{2}}\ll_{q}1.$

Hence,

(25) $\displaystyle\int_{0}^{x}\left|\sum_{\mid\gamma_{\chi}\mid\leq
T}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}\right|^{2}dt\ll_{q}x^{2B^{*}_{q}+1}.$

By substituting (25) into (22), we obtain

$H_{a,q}(x)\ll_{q}x^{2B^{*}_{q}+1}+\dfrac{x^{3}}{T^{2}}\log^{4}(qx).$

Now, we choose $T=x$. Therefore, since $B^{*}_{q}\geq\dfrac{1}{2}$ we obtain

$H_{a,q}(x)\ll_{q}x^{2B^{*}_{q}+1}+x\log^{4}(qx)\ll_{q}x^{2B^{*}_{q}+1}.$

∎

######  Lemma 3.5.

For $1\leq h\leq x$ and $x\gg_{q}1$, we have

$K_{a,q}(x,h)\ll_{q}hx^{2B_{q}^{*}}\log^{2}(qx).$

###### Proof.

We consider two cases:

Case 1:  Let GRH be true.

Prachar [18] showed that for $x\geq 2,1\leq q\leq x,1\leq h\leq x$, we have

$\underset{a(q)}{\sum}^{*}K_{a,q}(x,h)\ll hx\log^{2}(qx),$

where $\sum_{a(q)}^{*}$ denotes a sum over a set of reduced residues modulo
$q$. Thus this lemma is true under the GRH.

Case 2: Let GRH be false, which means that $B_{q}>1/2$.

For $0<\vartheta\leq 1$, we consider

$\displaystyle K(x,\vartheta)$
$\displaystyle=\int_{x}^{2x}\left(\psi(t+t\vartheta,q,a)-\psi(t,q,a)-\dfrac{t\vartheta}{\varphi(q)}\right)^{2}dt$
$\displaystyle\ll\int_{1}^{2}\int_{xv/2}^{2xv}\left(\psi(t+t\vartheta,q,a)-\psi(t,q,a)-\dfrac{t\vartheta}{\varphi(q)}\right)^{2}dtdv.$

Using (9) with $T=t$, we have

(26)
$\displaystyle\begin{split}\int_{xv/2}^{2xv}\left(\psi(t+t\vartheta,q,a)-\psi(t,q,a)-\dfrac{t\vartheta}{\varphi(q)}\right)^{2}dt&\ll_{q}\int_{xv/2}^{2xv}\left|\sum_{\chi(q)}\overline{\chi}(a)\sum_{\rho_{\chi}}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}((1+\vartheta)^{\rho_{\chi}}-1)\right|^{2}dt\\\
&+\int_{xv/2}^{2xv}\left(\log^{2}(qt(1+\vartheta)-\log^{2}(qt)\right)^{2}dt\\\
&=K_{1}(v)+K_{2}(v).\end{split}$

Trivially, $K_{2}(v)\ll xv\log^{4}(qx),$ then

(27) $\displaystyle\int_{1}^{2}K_{2}(v)dv\ll x\log^{4}(qx).$

By the Cauchy-Schwarz inequality, we obtain

$\displaystyle K_{1}(v)$
$\displaystyle\ll_{q}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\int_{xv/2}^{2xv}\left|\sum_{\rho_{\chi}}\dfrac{t^{\rho_{\chi}}}{\rho_{\chi}}((1+\vartheta)^{\rho_{\chi}}-1)\right|^{2}dt$
$\displaystyle=\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\int_{xv/2}^{2xv}\sum_{\rho_{\chi}}\sum_{\rho^{\prime}_{\chi}}\dfrac{(1+\vartheta)^{\rho_{\chi}}-1}{\rho_{\chi}}.\dfrac{(1+\vartheta)^{\overline{\rho^{\prime}_{\chi}}}-1}{\overline{\rho^{\prime}_{\chi}}}t^{\rho_{\chi}}t^{\overline{\rho^{\prime}_{\chi}}}dt$
$\displaystyle=\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\sum_{\rho_{\chi}}\sum_{\rho^{\prime}_{\chi}}\dfrac{(1+\vartheta)^{\rho_{\chi}}-1}{\rho_{\chi}}.\dfrac{(1+\vartheta)^{\overline{\rho^{\prime}_{\chi}}}-1}{\overline{\rho^{\prime}_{\chi}}}\dfrac{2^{1+\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}}-2^{-1-\rho_{\chi}-\overline{\rho^{\prime}_{\chi}}}}{\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1}(xv)^{\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1}.$

Thus,

$\displaystyle\int_{1}^{2}K_{1}(v)dv\ll_{q}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\sum_{\rho_{\chi}}\sum_{\rho^{\prime}_{\chi}}\dfrac{(1+\vartheta)^{\rho_{\chi}}-1}{\rho_{\chi}}.\dfrac{(1+\vartheta)^{\overline{\rho^{\prime}_{\chi}}}-1}{\overline{\rho^{\prime}_{\chi}}}$
$\displaystyle.\dfrac{2^{1+\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}}-2^{-1-\rho_{\chi}-\overline{\rho^{\prime}_{\chi}}}}{\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1}$
$\displaystyle.\dfrac{2^{2+\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}}-1}{\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+2}x^{\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1}.$

By the trivial inequality $|z_{1}z_{2}|\ll|z_{1}|^{2}+|z_{2}|^{2}$,

$\displaystyle\int_{1}^{2}K_{1}(v)dv$
$\displaystyle\ll_{q}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\sum_{\rho_{\chi}}\sum_{\rho^{\prime}_{\chi}}\left|\dfrac{(1+\vartheta)^{\rho_{\chi}}-1}{\rho_{\chi}}.\dfrac{(1+\vartheta)^{\overline{\rho^{\prime}_{\chi}}}-1}{\overline{\rho^{\prime}_{\chi}}}\right|\dfrac{x^{\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1}}{\left|\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1\right|^{2}}$
$\displaystyle\ll
x^{2B_{q}^{*}+1}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\sum_{\rho_{\chi}}\left|\dfrac{(1+\vartheta)^{\rho_{\chi}}-1}{\rho_{\chi}}\right|^{2}\sum_{\rho^{\prime}_{\chi}}\dfrac{1}{\left|\rho_{\chi}+\overline{\rho^{\prime}_{\chi}}+1\right|^{2}}$
$\displaystyle\ll
x^{2B_{q}^{*}+1}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\sum_{\gamma_{\chi}}\min\\{\vartheta^{2},(\gamma_{\chi})^{-2}\\}\sum_{\gamma^{\prime}_{\chi}}\dfrac{1}{(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)^{2}}.$

Using (23), we can estimate

$\displaystyle\sum_{\gamma^{\prime}_{\chi}}\dfrac{1}{(1+|\gamma_{\chi}-\gamma^{\prime}_{\chi}|)^{2}}\ll\log(q\gamma_{\chi}).$

From the above,

(28)
$\displaystyle\begin{split}\int_{1}^{2}K_{1}(v)dv&\ll_{q}x^{2B_{q}^{*}+1}\sum_{\chi(q)}|\overline{\chi}(a)|^{2}\left(\sum_{\gamma_{\chi}>\vartheta^{-1}}\dfrac{\log(q\gamma_{\chi})}{\gamma_{\chi}^{2}}+\sum_{0<\gamma_{\chi}\leq\vartheta^{-1}}\vartheta^{2}\log(q\gamma_{\chi})\right)\\\
&\ll_{q}x^{2B_{q}^{*}+1}\vartheta\log^{2}\left(\dfrac{1}{\vartheta}\right).\end{split}$

From (26), (27) and (28), we obtain

$\displaystyle
K(x,\vartheta)\ll_{q}x^{2B_{q}^{*}+1}\vartheta\log^{2}\left(\dfrac{1}{\vartheta}\right)+x\log^{4}(qx).$

Similar to [20, (6.21)], we have

$\displaystyle\int_{x}^{2x}$
$\displaystyle\left(\psi(t+h,q,a)-\psi(t,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt$
$\displaystyle\ll\dfrac{x}{h}\int_{h/3x}^{3h/x}\left(\int_{x}^{3x}\left(\psi(t+t\vartheta,q,a)-\psi(t,q,a)-\dfrac{t\vartheta}{\varphi(q)}\right)^{2}dt\right)d\vartheta$
$\displaystyle\ll_{q}hx^{2B_{q}^{*}}\log^{2}\left(\dfrac{x}{h}\right)+x\log^{4}(qx).$

Since $B_{q}^{*}>1/2$, there exits $\varepsilon(q)>0$ such that
$2B_{q}^{*}>1+\varepsilon(q)$. Thus,

$\displaystyle x\log^{4}(qx)\ll x^{2B_{q}^{*}-\varepsilon(q)}\log^{4}(qx)\ll
x^{2B_{q}^{*}}\log^{2}(qx)\text{ for }x\gg_{q}1.$

Therefore

(29) $\displaystyle\int_{x}^{2x}$
$\displaystyle\left(\psi(t+h,q,a)-\psi(t,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt\ll_{q}hx^{2B_{q}^{*}}\log^{2}(qx).$

Now we split $K_{a,q}(x,h)$ into two parts.

$\displaystyle K_{a,q}(x,h)$
$\displaystyle=\int_{0}^{h}\left(\psi(t+h,q,a)-\psi(t,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt$
$\displaystyle+\int_{h}^{x}\left(\psi(t+h,q,a)-\psi(t,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt.$

The first is estimated as

$\displaystyle\ll\int_{0}^{h}\left(\psi(t+h,q,a)-\dfrac{t+h}{\varphi(q)}\right)^{2}dt+\int_{0}^{h}\left(\psi(t,q,a)-\dfrac{t}{\varphi(q)}\right)^{2}dt$
$\displaystyle\ll_{q}h^{2B_{q}^{*}+1}.$

We have $[h,x]$ is recovered by the disjoint union of
$[\frac{x}{2^{k+1}},\frac{x}{2^{k}}]$, for $0\leq
k\leq\mathcal{O}(\log\dfrac{x}{h})$. Using (29), we rewrite the second part as

$\displaystyle\ll\sum_{k\geq
0}\int_{x/{2^{k+1}}}^{x/{2^{k}}}\left(\psi(t+h,q,a)-\psi(t,q,a)-\dfrac{h}{\varphi(q)}\right)^{2}dt$
$\displaystyle\ll_{q}h\sum_{k\geq
0}\left(\dfrac{x}{2^{k}}\right)^{2B_{q}^{*}}\log^{2}(qx)$
$\displaystyle\ll_{q}hx^{2B_{q}^{*}}\log^{2}(qx).$

Combining two estimate above, the proof is complete. ∎

Now, using Lemmas 3.2, 3.3, 3.4, 3.5 and (14), we complete the proof of part
(A) Theorem 1.3.

$\displaystyle W(X,q,a,h)\ll$
$\displaystyle\dfrac{1}{h^{2}}\left(h^{2B_{q}^{*}+1}+\dfrac{h}{\varphi^{2}(q)}\right)$
$\displaystyle+\dfrac{1}{X^{2}}\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left((jX)^{2B_{q}^{*}+1}+\dfrac{jX}{\varphi^{2}(q)}\right)$
$\displaystyle+\dfrac{1}{h^{2}}\sum_{j=1}^{\infty}\dfrac{1}{2^{j-1}}\left(h(jX)^{2B_{q}^{*}}\log^{2}(qjX)+\dfrac{jX}{\varphi^{2}(q)}\right)$
$\displaystyle\ll_{q}$ $\displaystyle
h^{2B^{*}_{q}-1}+X^{2B^{*}_{q}-1}\sum_{j\geq
1}\dfrac{j^{2B^{*}_{q}+1}}{2^{j-1}}+\dfrac{X^{2B^{*}_{q}}}{h}\sum_{j\geq
1}\dfrac{j^{2B^{*}_{q}}}{2^{j-1}}\log^{2}(qX)$ $\displaystyle\ll_{q}$
$\displaystyle\dfrac{X^{2B^{*}_{q}}}{h}\log^{2}(qX).$

Substituting the above inequality into (13), we get

$\displaystyle T_{1}(X,q,a)$ $\displaystyle\ll_{q}X^{2B^{*}_{q}}\log
X\log^{2}(qX).$

We conclude

$\displaystyle E(X,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle\ll_{q_{1},q_{2}}T_{1}(X,q_{1},a_{1})^{1/2}T_{1}(X,q_{2},a_{2})^{1/2}$
$\displaystyle\ll_{q_{1},q_{2}}X^{B^{*}_{q_{1}}+B^{*}_{q_{2}}}\log
X\log(q_{1}X)\log(q_{2}X),$

##  4\. Proof of Theorem 1.9

First, let

(30) $\displaystyle\mathcal{K}(X,z):=\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)z^{n}.$

Then we have

$\displaystyle\int^{1}_{0}\left(F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right)\left(F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right)\mathcal{K}\left(X,\dfrac{1}{z}\right)d\alpha$
$\displaystyle=\int^{1}_{0}\sum_{n}B(n,q_{1},q_{2},a_{1},a_{2})r^{n}e(n\alpha)\sum_{m\leq
X}\left(1-\dfrac{m}{X}\right)r^{-m}e(-m\alpha)d\alpha$
$\displaystyle=\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)B(n,q_{1},q_{2},a_{1},a_{2}).$

From (6) and Lemma 2.1, we have

$\displaystyle\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)B(n,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle=\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)G(n,q_{1},q_{2},a_{1},a_{2})-\dfrac{1}{\varphi(q_{1})X}\int_{1}^{X}\sum_{n\leq
u}\psi(n-1,q_{2},a_{2})du$
$\displaystyle-\dfrac{1}{\varphi(q_{2})X}\int_{1}^{X}\sum_{n\leq
u}\psi(n-1,q_{1},a_{1})du+\dfrac{1}{\varphi(q_{1})\varphi(q_{2})X}\int_{1}^{X}\sum_{n\leq
u}(n-1)du$ $\displaystyle=\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)G(n,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle-\dfrac{1}{\varphi(q_{1})\varphi(q_{2})X}\int_{1}^{X}\left(\dfrac{u^{2}}{2}-\sum_{\chi(q_{1})}\overline{\chi}(a_{1})\sum_{\rho_{\chi}}\dfrac{u^{\rho_{\chi}+1}}{{\rho_{\chi}}({\rho_{\chi}}+1)}+\mathcal{O}(u\log^{2}(q_{1}u))\right)$
$\displaystyle-\left(\sum_{\chi(q_{2})}\overline{\chi}(a_{2})\sum_{\rho_{\chi}}\dfrac{u^{\rho_{\chi}+1}}{{\rho_{\chi}}({\rho_{\chi}}+1)}+\mathcal{O}(u\log^{2}(q_{2}u))\right)du.$

Then we obtain

$\displaystyle\sum_{n\leq
X}\left(1-\dfrac{n}{X}\right)G(n,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle=\dfrac{X^{2}}{6\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}L(X,q_{1},a_{1})+\dfrac{1}{\varphi(q_{1})}L(X,q_{2},a_{2})$
$\displaystyle+\mathcal{O}(X(\log^{2}(q_{1}X)+\log^{2}(q_{2}X))+\mathcal{E}(X,q_{1},q_{2},a_{1},a_{2}),$

where

$\mathcal{E}(X,q_{1},q_{2},a_{1},a_{2})=\int^{1}_{0}\left(F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right)\left(F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right)\mathcal{K}\left(X,\dfrac{1}{z}\right)d\alpha.$

By the Cauchy-Schwarz inequality, we have

$\displaystyle\mid\mathcal{E}(X,q_{1},q_{2},a_{1},a_{2})\mid^{2}\ll$
$\displaystyle\int^{1}_{0}\left|F_{a_{1},q_{1}}(z)-I_{q_{1}}(z)\right|^{2}\left|\mathcal{K}\left(X,\dfrac{1}{z}\right)\right|d\alpha$
$\displaystyle\times$
$\displaystyle\int^{1}_{0}\left|F_{a_{2},q_{2}}(z)-I_{q_{2}}(z)\right|^{2}\left|\mathcal{K}\left(X,\dfrac{1}{z}\right)\right|d\alpha.$

We need only estimate

$T_{2}(X,q,a):=\int^{1}_{0}\left|F_{a,q}(z)-I(z)\right|^{2}\left|\mathcal{K}\left(X,\dfrac{1}{z}\right)\right|d\alpha.$

We consider

$\displaystyle\mathcal{K}\left(X,\dfrac{1}{z}\right)$
$\displaystyle=\sum_{n\leq
x}\left(1-\dfrac{n}{X}\right)e^{n/X}e(-n\alpha)=\dfrac{1}{X}\int^{X}_{1}\sum_{n\leq
t}e^{n/X}e(-n\alpha)dt$
$\displaystyle=\dfrac{1}{X}\int^{X}_{1}I\left(t,\dfrac{1}{z}\right)dt$
$\displaystyle\ll\min\left(X,\dfrac{1}{|\alpha|}\right),$

since (11). Using Lemma 3.1 and similar to the proof of the part (A) Theorem
1.3, we also obtain

$\displaystyle T_{2}(X,q,a)$ $\displaystyle\ll_{q}X^{2B^{*}_{q}}\log
X\log^{2}(qX).$

We conclude

$\displaystyle\mathcal{E}(X,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle\ll_{q_{1},q_{2}}X^{B^{*}_{q_{1}}+B^{*}_{q_{2}}}\log
X\log(q_{1}X)\log(q_{2}X).$

##  5\. Proof of the part (B) Theorem 1.3

By the part (A), we obtain the main term in this formula. We will show that

(31) $\displaystyle G(n,q_{1},q_{2},a_{1},a_{2})=\Omega(n\log\log n),$

and then $(B)$ is proved. In fact that, assuming the error term of $(B)$ is
$o(X\log\log X)$ then

$\displaystyle S(n+1,q_{1},q_{2},a_{1},a_{2})-S(n,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle=\dfrac{(n+1)^{2}-n^{2}}{2\varphi(q_{1})\varphi(q_{2})}+\dfrac{1}{\varphi(q_{2})}\left(H(n+1,q_{1},a_{1})-H(n,q_{1},a_{1})\right)$
$\displaystyle+\dfrac{1}{\varphi(q_{1})}\left(H(n+1,q_{2},a_{2})-H(n,q_{2},a_{2})\right)+o(n\log\log
n)$ $\displaystyle=o(n\log\log n).$

So (31) would be false. Hence the error term of $(B)$ is $\Omega(X\log\log
X)$.

We recall the following result.

######  Lemma 5.1.

Let $q^{\prime}$ be a positive integer, $q=\prod_{\begin{subarray}{c}p\leq
x\\\ p\neq p_{1},p\nmid q^{\prime}\end{subarray}}p$, where $p_{1}$ is some
prime divisor of the exceptional modulus up to $q$ if there exists a Siegel’s
zero, then

$\displaystyle\left|x-\sum_{x\leq n\leq
2x}\Lambda(n)\chi_{0}(n)\right|+\sum_{\begin{subarray}{c}\chi(q)\\\
\chi\neq\chi_{0}\end{subarray}}\left|\sum_{x\leq n\leq
2x}\Lambda(n)\chi(n)\right|\leq\dfrac{x}{2}.$

This lemma is implied by [8, Theorem 7], we also see it in [2, Lemma 4]. It
follows that, for $(a,q)=1$,

$\displaystyle\psi(2x,q,a)=\sum_{\begin{subarray}{c}n\leq 2x\\\ n\equiv
a(q)\end{subarray}}\Lambda(n)\geq\sum_{\begin{subarray}{c}x\leq n\leq 2x\\\
n\equiv a(q)\end{subarray}}\Lambda(n)$
$\displaystyle=\dfrac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi}(a)\sum_{x\leq
n\leq 2x}\chi(n)\Lambda(n)$ $\displaystyle=\dfrac{1}{\varphi(q)}\sum_{x\leq
n\leq
2x}\Lambda(n)\chi_{0}(n)+\dfrac{1}{\varphi(q)}\sum_{\begin{subarray}{c}\chi(q)\\\
\chi\neq\chi_{0}\end{subarray}}\overline{\chi}(a)\sum_{x\leq n\leq
2x}\chi(n)\Lambda(n)$
$\displaystyle=\dfrac{x}{\varphi(q)}-\dfrac{1}{\varphi(q)}\left(x-\sum_{x\leq
n\leq 2x}\Lambda(n)\chi_{0}(n)-\sum_{\begin{subarray}{c}\chi(q)\\\
\chi\neq\chi_{0}\end{subarray}}\overline{\chi}(a)\sum_{x\leq n\leq
2x}\Lambda(n)\chi(n)\right)$
$\displaystyle\geq\dfrac{x}{\varphi(q)}-\dfrac{1}{\varphi(q)}\left(\left|x-\sum_{x\leq
n\leq 2x}\Lambda(n)\chi_{0}(n)\right|+\sum_{\begin{subarray}{c}\chi(q)\\\
\chi\neq\chi_{0}\end{subarray}}\left|\overline{\chi}(a)\right|\left|\sum_{x\leq
n\leq 2x}\Lambda(n)\chi(n)\right|\right)$
$\displaystyle\geq\dfrac{x}{\varphi(q)}-\dfrac{x}{2\varphi(q)}=\dfrac{x}{2\varphi(q)}.$

Let us now

$Q=\prod_{\begin{subarray}{c}p\leq x\\\ p\neq p_{1},p\nmid
q_{1}q_{2}\end{subarray}}p.$

Hence,

(32) $\displaystyle\sum_{\begin{subarray}{c}n\leq 4x\\\ Q\mid
n\end{subarray}}G(n,q_{1},q_{2},a_{1},a_{2})\geq\sum_{\begin{subarray}{c}b=1\\\
(b,Q)=1\end{subarray}}^{Q}\sum_{\begin{subarray}{c}m\leq 2x\\\ m\equiv
a_{1}(q_{1})\\\ m\equiv
b(Q)\end{subarray}}\Lambda(m)\sum_{\begin{subarray}{c}l\leq 2x\\\ l\equiv
a_{2}(q_{2})\\\ l\equiv Q-b(Q)\end{subarray}}\Lambda(l).$

We use the Chinese remainder Theorem to evaluate the right hand side of (32).
Since $Q$ and $q_{1}$ are coprime, the solutions of the system $m\equiv
a_{1}(q_{1})$ and $m\equiv b(Q)$ are given by

$m\equiv a_{1}m_{1}Q+bm_{2}q_{1}(q_{1}Q):=A(q_{1}Q),$

where $m_{1},m_{2}$ are integers satisfying $m_{1}Q+m_{2}q_{1}=1$.

Similar to the system $l\equiv a_{2}(q_{2})$ and $l\equiv Q-b(Q)$, we have

$l=:B(q_{2}Q).$

Thus, using Lemma 5.1, the right hand side of (32) is

$\displaystyle\sum_{\begin{subarray}{c}b=1\\\
(b,Q)=1\end{subarray}}^{Q}\sum_{\begin{subarray}{c}m\leq 2x\\\ m\equiv
A(q_{1}Q)\end{subarray}}\Lambda(m)\sum_{\begin{subarray}{c}l\leq 2x\\\ l\equiv
B(q_{2}Q)\end{subarray}}\Lambda(l)=\sum_{\begin{subarray}{c}b=1\\\
(b,Q)=1\end{subarray}}^{Q}\psi(2x,q_{1}Q,A)\psi(2x,q_{2}Q,B)$
$\displaystyle\geq\sum_{\begin{subarray}{c}b=1\\\
(b,Q)=1\end{subarray}}^{Q}\dfrac{x}{2\varphi(q_{1}Q)}\dfrac{x}{2\varphi(q_{2}Q)}=\dfrac{x^{2}}{4\varphi(q_{1})\varphi(q_{2})\varphi(Q)},$

since $(q_{1},Q)=1$, $(q_{2},Q)=1$.

Hence, we obtain

$\displaystyle\max_{n\leq 4x}G(n,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle\geq\dfrac{Q}{4x}\sum_{\begin{subarray}{c}n\leq 4x\\\ Q\mid
n\end{subarray}}G(n,q_{1},q_{2},a_{1},a_{2})$
$\displaystyle\geq\dfrac{xQ}{16\varphi(q_{1})\varphi(q_{2})\varphi(Q)}=\dfrac{x}{16\varphi(q_{1})\varphi(q_{2})}\prod_{\begin{subarray}{c}p\leq
x\\\ p\neq p_{1},p\nmid q_{1}q_{2}\end{subarray}}(1-p^{-1})^{-1}$
$\displaystyle=(1-p_{1}^{-1})\dfrac{\prod_{p\mid
q_{1}q_{2}}(1-p^{-1})}{16\varphi(q_{1})\varphi(q_{2})}x\prod_{p\leq
x}(1-p^{-1})^{-1}$ $\displaystyle\gg_{q_{1},q_{2}}x\log\log x.$

## Acknowledgement

I would like to thank Didier Lesesvre for inspiring discussions on the topics
of this paper. I would also like to thank Alessandro Languasco for the his
careful reading and his comments. I am grateful to Gautami Bhowmik for her
guidance together with many useful comments and suggestions during the
preparation of this paper.

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