On the divisor problem with congruence conditions

TL;DR

This paper investigates the behavior of the error term in a divisor problem with congruence conditions, proving sign change results and the existence of large fluctuations in short intervals.

Contribution

It establishes sign change intervals and large fluctuation results for the error term in a divisor problem with congruence conditions, extending understanding of its oscillatory nature.

Findings

Sign changes occur within intervals of length proportional to ^{1/2}.

Infinitely many short intervals exhibit large deviations of the error term.

The error term's magnitude can be bounded below by a constant times x^{1/4} in certain subintervals.

Abstract

Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\equiv r_i\pmod{q_i}$ ($i=1,2$) and $\Delta(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with $x\geq (q_1q_2)^{1+\varepsilon}, 1\leq r_i\leq q_i$, and $(r_i, q_i)=1$ ($i=1, 2$). We study the power moments and sign changes of $\Delta(x; r_1, q_1, r_2, q_2)$, and prove that for a sufficiently large constant $C$, $\Delta(q_1q_2x; r_1, q_1, r_2, q_2)$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c'$, there exist infinitely many subintervals of length $c'\sqrt{T}\log^{-7}T$ in $[T,2T]$ where $\pm \Delta(q_1q_2x; r_1, q_1, r_2, q_2)> c_5x^\frac{1}{4}$ always holds.

Equations

$$D(x)=x\log x+(2\gamma -1)x+O(\sqrt{x}),$$

$$\Delta (x)=D(x)-x\log x-(2\gamma -1)x$$

$$\Delta (x)\ll x^{\frac{131}{416}}{\log }^{\frac{26947}{8320}}x.$$

$${\int }_1^T\Delta (x)dx=\frac{T}{4}+O(T^{\frac{3}{4}}).$$

$${\int }_1^T\Delta (x{)}^{2}dx=c{T}^{\frac{3}{2}}+O(T^{\frac{5}{4}+\epsilon }),\forall \epsilon >0,$$

$${\int }_1^T|\Delta (x){|}^{A}dx\ll T^{1+\frac{A}{4}+\epsilon },\forall \epsilon >0$$

$${\int }_1^T\Delta (x{)}^{k}dx=c_k{T}^{1+\frac{k}{4}}+O(T^{1+\frac{k}{4}-{\delta }_k}),\text{for}k=3,4,$$

$$c_k=\underset{X\to \infty }{\lim }{X}^{-1-\frac{k}{4}}{\int }_1^X\Delta (x{)}^{k}dx$$

$$\Delta(x)=-\frac{2}{\pi}\sqrt{x}\sum_{n=1}^{\infty}\frac{d(n)}{\sqrt{n}}\big{(}K_{1}(4\pi\sqrt{nx})+\frac{\pi}{2}Y_{1}(4\pi\sqrt{nx})\big{)},$$

$$d(n;r_1,q_1,r_2,q_2)={\sum }_{\begin{array}{c}n=n_1{n}_2\\ n_i\equiv r_i(\mathrm{mod}{q}_i)\\ i=1,2\end{array}}1,$$

$$D(x;r_1,q_1,r_2,q_2)={\sum }_{\begin{array}{c}{n}_1{n}_2\le x\\ n_i\equiv r_i(\mathrm{mod}{q}_i)\\ i=1,2\end{array}}1.$$

$$D(x;r_{1},q_{1},r_{2},q_{2})\\ =\frac{x}{q_{1}q_{2}}\log\Big{(}\frac{x}{q_{1}q_{2}}\Big{)}-\bigg{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{r_{1}}{q_{1}}\Big{)}+\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{r_{2}}{q_{2}}\Big{)}+1\bigg{)}\frac{x}{q_{1}q_{2}}+\Delta(x;r_{1},q_{1},r_{2},q_{2}).$$

$$\Delta(x;r_{1},q_{1},r_{2},q_{2})\ll\Big{(}\frac{x}{q_{1}q_{2}}\Big{)}^{\frac{131}{416}}\Big{(}\log\Big{(}\frac{x}{q_{1}q_{2}}\Big{)}\Big{)}^{\frac{26947}{8320}}$$

$$\Delta(x;r_{1},q_{1},r_{2},q_{2})\ll\Big{(}\frac{x}{q_{1}q_{2}}\Big{)}^{\frac{1}{4}+\varepsilon}$$

$$\int_{1}^{T}{\Delta(x;r_{1},q_{1},r_{2},q_{2})}dx=\big{(}\frac{r_{1}}{q_{1}}-\frac{1}{2}\big{)}\big{(}\frac{r_{2}}{q_{2}}-\frac{1}{2}\big{)}T+O\big{(}(q_{1}q_{2})^{\frac{1}{4}}T^{\frac{3}{4}}\big{)},$$

$$\int_{1}^{T}{\Delta^{2}(x;r_{1},q_{1},r_{2},q_{2})}dx=c_{2}(q_{1}q_{2})^{\frac{1}{2}}T^{\frac{3}{2}}+o\big{(}(q_{1}q_{2})^{\frac{1}{2}}T^{\frac{3}{2}}\big{)},$$

$${\int }_1^T|\Delta (q_1{q}_{2}x;r_1,q_1,r_2,q_2){|}^{A}dx\ll T^{1+\frac{A}{4}}{\mathcal{L}}^{4A},$$

$${\int }_1^T|\Delta (q_1{q}_{2}x;r_1,q_1,r_2,q_2){|}^{A_0}dx\ll T^{1+\frac{A_0}{4}+\epsilon },$$

$$\int_{1}^{T}{\Delta^{k}(q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2})}dx=C_{k}\int_{1}^{T}x^{\frac{k}{4}}dx+o\big{(}T^{1+\frac{k}{4}}\big{)},$$

$$meas\big{\{}t\in[T,2T]:\pm\Delta(q_{1}q_{2}t;r_{1},q_{1},r_{2},q_{2})>c_{5}t^{\frac{1}{4}}\big{\}}\gg T.$$

$$\mathcal{F}_{k}\big{(}q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2}\big{)}:=\int_{1}^{T}{\Delta^{k}\big{(}q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2}\big{)}}dx-C_{k}T^{1+\frac{k}{4}}.$$

$$\mathcal{F}_{k}\big{(}q_{1}q_{2}X;r_{1},q_{1},r_{2},q_{2}\big{)}\gg X^{\frac{1}{2}+\frac{k}{4}}\mathcal{L}^{-7}.$$

$$\Delta(q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2})=R_{0}(x;y)+{R_{12}}(x;y,H)+{R_{21}}(x;y,H)\\ +G_{12}(x;H)+G_{21}(x;H)+O\big{(}\log^{3}(q_{1}q_{2}T)\big{)},$$

$$R_0(x;y)$$

$$R_{12}(x;y,H)$$

$$R_{21}(x;y,H)$$

$$G_{12}(x;H)$$

$$G_{21}(x;H)$$

$$\Delta^{**}(t)=\sqrt{2}\pi t^{-\frac{1}{2}}\Big{(}\Delta(q_{1}q_{2}t^{2};r_{1},q_{1},r_{2},q_{2})+f(t^{2})\Big{)},\quad\text{for }t\geq 1.$$

$$K_{\zeta}(u):=(1-|u|)\big{(}1+\zeta\sin(4\pi\alpha u)\big{)}\quad\text{for }|u|\leq 1,$$

$${\int }_{-1}^1{\Delta }^{\ast \ast }(t+\alpha u)K_{\zeta }(u)du$$

$$=$$

$$\displaystyle+O\big{(}t^{-\frac{1}{2}}\sup_{|u|\leq 1}f((t\!+\!\alpha u)^{2})\big{)}\!+\!O\big{(}t^{-\frac{1}{2}}\mathcal{L}^{3}\big{)}.$$

Full text

The first author is supported by the National Natural Science Foundation of China (Grant No. 11871295 and Grant No. 11571303), China Postdoctoral Science Foundation (Grant No. 2018M631434). The second author is supported by the National Key Basic Research Program of China (Grant No. 2013CB834201). The third author is supported by the National Natural Science Foundation of China (Grant No. 11571303).

1. Introduction

1.1. Dirichlet divisor problem

Let $d(n)$ be the Dirichlet divisor function, $D(x)=\mathop{{\sum}}\limits_{n\leq x}d(n)=\mathop{{\sum}}\limits_{n_{1}n_{2}\leq x}1$ be the summatory function. In 1849, Dirichlet proved that

[TABLE]

where $\gamma$ is the Euler constant.

Let

[TABLE]

be the error term in the asymptotic formula for $D(x)$. Dirichlet’s divisor problem consists of determining the smallest $\alpha$, for which $\Delta(x)\ll x^{\alpha+\varepsilon}$ holds for any $\varepsilon>0$. Clearly, Dirichlet’s result implies that $\alpha\leq\frac{1}{2}$. Since then, there are many improvements on this estimate. The best to-date is given by Huxley[5, 6], reads

[TABLE]

It is widely conjectured that $\alpha=\frac{1}{4}$ is admissible and is the best possible.

Since $\Delta(x)$ exhibits considerable fluctuations, one natural way to study the upper bounds is to consider the moments.

In 1904, Voronoi [17] showed that

[TABLE]

Later, in 1922 Cramér[1] proved the mean square formula

[TABLE]

where $c$ is a positive constant. In 1983, Ivic [7] used the method of large values to prove that

[TABLE]

for each fixed $0\leq A\leq\frac{35}{4}$. The range of $A$ can be extended to $\frac{262}{27}$ by the estimate (1.1). In 1992, Tsang[15] obtained the asymptotic formula

[TABLE]

with positive constants $c_{3}$, $c_{4}$, and $\delta_{3}=\frac{1}{14}$, $\delta_{4}=\frac{1}{23}$. Ivić and Sargos [8] improved the values $\delta_{3}$, $\delta_{4}$ to $\delta^{\prime}_{3}=\frac{7}{20}$, $\delta^{\prime}_{4}=\frac{1}{12}$, respectively. Heath-Brown[3] in 1992 proved that for any positive real number $k<A$, where $A$ satisfies (1.2), the limit

[TABLE]

exists. Then, there followed a series of investigations on explicit asymptotic formula of the type (1.3) for larger values of $k$. In 2004, Zhai [18] established asymptotic formulas for $3\leq k\leq 9$.

At the beginning of the 20th century, Voronoi[17] proved the remarkable exact formula that

[TABLE]

where $K_{1}$, $Y_{1}$ are the Bessel functions, and the series on the right-hand side is boundedly convergent for $x$ lying in each fixed closed interval.

Heath-Brown and Tsang [4] studied the sign changes of $\Delta(x)$. They proved that for a suitable constant $C>0$, $\Delta(x)$ changes sign on the interval $[T,T+C\sqrt{T}]$ for every sufficiently large $T$. Here the length $\sqrt{T}$ is almost best possible since they proved that in the interval $[T,2T]$ there are many subintervals of length $\gg\sqrt{T}\log^{-5}T$ such that $\Delta(x)$ does not change sign in any of these subintervals.

1.2. The divisor problem with congruence conditions

A divisor function with congruence conditions is defined by

[TABLE]

of which, the summatory function is

[TABLE]

From Richert [13], we can find that for $x\geq q_{1}q_{2}$, $1\leq r_{i}\leq q_{i}$ ($i=1,2$)

[TABLE]

From Huxley’s estimates[5], it follows that

[TABLE]

uniformly in $1\leq r_{1}\leq q_{1}\leq x,1\leq r_{2}\leq q_{2}\leq x$. It is conjectured that

[TABLE]

uniformly in $1\leq r_{1}\leq q_{1}\leq x,1\leq r_{2}\leq q_{2}\leq x$, $\forall\varepsilon>0$, which is an analogue of the well-known conjecture that $\Delta(x)\ll x^{\frac{1}{4}+\varepsilon}$.

Müller and Nowak[12] studied the mean value of $\Delta(x;r_{1},q_{1},r_{2},q_{2})$. They pointed out

[TABLE]

and

[TABLE]

uniformly in $1\leq r_{i}\leq q_{i}\leq T$ $(i=1,2)$, if $T$ is a large number, and $c_{2}$ is a constant.

In [9], we show that

[TABLE]

for $0\leq A\leq\frac{262}{27}$ and $T\gg(q_{1}q_{2})^{\varepsilon}$.

Here we study $\Delta(x;r_{1},q_{1},r_{2},q_{2})$ further and give some more results about it.

Notations. For a real number $t$, let $[t]$ be the largest integer no greater than $t$, $\{t\}=t-[t]$, $\psi(t)=\{t\}-\frac{1}{2}$, $\parallel t\parallel=\min(\{t\}$, $1-\{t\})$, $e(t)=e^{2\pi it}$. $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{N}$ denote the set of complex numbers, of real numbers, of integers, and of natural numbers, respectively; $f\asymp g$ means that both $f\ll g$ and $f\gg g$ hold. Throughout this paper, $\varepsilon$ denote sufficiently small positive constants, and $\mathcal{L}$ denotes $\log T$.

2. Main results

In this paper, we will first discuss the power moments of $\Delta(x;r_{1},q_{1},r_{2},q_{2})$ and get the following

Theorem 2.1**.**

If $T\gg(q_{1}q_{2})^{\varepsilon}$ is large enough. If $A_{0}>9$ satisfies

[TABLE]

then for any fixed integer $3\leq k<A_{0}$, we have

[TABLE]

where $C_{k}\asymp 1$ are explicit constants.

From (1.9), we can take $A_{0}=\frac{262}{27}$, which means

Corollary 2.1**.**

If $T,r_{i}$ and $q_{i}(i=1,2)$ satisfying the hypothesis of Theorem 2.1, then (2.1) holds for any fixed integer $3\leq k\leq 9$.

By using the estimates above, we can get the sign changes of $\Delta(x;r_{1},q_{1},r_{2},q_{2})$ as following

Theorem 2.2**.**

Let $c_{1}>0$ be a sufficiently small constant and $c_{2}>0$ be a sufficiently large constant, $q_{1}\geq 2$, $q_{2}\geq 3$, $1\leq r_{i}\leq q_{i}$ and $(r_{i},q_{i})=1$ $(i=1,2)$. For any real-valued function $|f(t)|\leq c_{1}t^{\frac{1}{4}}$, the function $\Delta(q_{1}q_{2}t;r_{1},q_{1},r_{2},q_{2})+f(t)$ changes sign at least once in the interval $[T,T+c_{2}\sqrt{T}]$ for every sufficiently large $T\gg(q_{1}q_{2})^{\varepsilon}$. In particular, there exist $t_{1}$, $t_{2}\in[T,T+c_{2}\sqrt{T}]$ such that $\Delta(q_{1}q_{2}t_{1};r_{1},q_{1},r_{2},q_{2})\geq c_{1}t_{1}^{\frac{1}{4}}$ and $\Delta(q_{1}q_{2}t_{2};r_{1},q_{1},r_{2},q_{2})\leq-c_{1}t_{2}^{\frac{1}{4}}$.

Theorem 2.3**.**

There exist three positive absolute constants $c_{3}$ ,$c_{4}$ ,$c_{5}$ such that, for any large parameter $T\gg(q_{1}q_{2})^{\varepsilon}$, and any choice of $\pm$ signs, there are at least $c_{3}\sqrt{T}log^{7}T$ disjoint subintervals of length $c_{4}\sqrt{T}log^{-7}T$ in $[T,2T]$, such that $\pm\Delta(q_{1}q_{2}t;r_{1},q_{1},r_{2},q_{2})>c_{5}t^{\frac{1}{4}}$, whenever $t$ lies in any of these subintervals. Moreover, we have the estimate

[TABLE]

We also study the $\Omega$-result of the error term in the asymptotic formula (2.1) for odd $k$ by using Theorem 2.3. Define

[TABLE]

We have the following

Theorem 2.4**.**

For any $T\gg(q_{1}q_{2})^{\varepsilon}$, the interval $[T,2T]$ contains a point $X$, for which

[TABLE]

Remark 2.1**.**

Although at the present moment we can only prove (2.1) for $2\leq k\leq 9$, Theorem 2.4 holds for any odd $k\geq 2$.

3. proof of Theorem 2.1

In this section, we prove Theorem 2.1 by using the Voronoi-type formula for $\Delta(x;r_{1},q_{1},r_{2},q_{2})$.

Lemma 3.1**.**

$($* See [9] $)$*

Let $J=[\frac{\mathcal{L}+2\log q_{1}q_{2}-4\log\mathcal{L}}{\log 2}]$, $H\geq 2$ be a parameter to be determined, and $T^{\varepsilon}<y\leq\min(H^{2},(q_{1}q_{2})^{2}T)\mathcal{L}^{-4}$. Suppose $\frac{T}{2}\leq x\leq T$. Then

[TABLE]

where

[TABLE]

where ${\sum\limits_{n\leq x}}^{\prime}f(n)$ indicates that if $x$ is an integer, then only $\frac{1}{2}f(x)$ is counted.

Thus, we can get Theorem 2.1 by using Lemma 3.1 with the approach of Liu [11].

4. Proof of Theorem 2.2

In this section, we prove Theorem 2.2 following the approach of [4].

Suppose $|f(t)|\leq c_{1}t^{\frac{1}{4}}$. Let

[TABLE]

Define

[TABLE]

with $\zeta=1$ or $-1$, and $\alpha>1$ a large number.

Lemma 4.1**.**

Suppose $T\gg(q_{1}q_{2})^{\varepsilon}$ is a large parameter. Then for each $\sqrt{T}\leq t\leq\sqrt{2T}$, we have

[TABLE]

Proof.

Let $J=[\frac{\mathcal{L}+2\log q_{1}q_{2}-4\log\mathcal{L}}{\log 2}]$, $H\geq 2$ be a parameter to be determined, and $T^{\varepsilon}<y\leq\min(H^{2},(q_{1}q_{2})^{2}T)\mathcal{L}^{-4}$. From (3.1), we have

[TABLE]

where

[TABLE]

Denote

[TABLE]

Then

[TABLE]

We first consider $\int_{-1}^{1}G^{*}(t+\alpha u)du$. Noting that

[TABLE]

with

[TABLE]

We have

[TABLE]

where the first derivative test was used. This estimate remain valid with $G^{*}_{12}$ replaced by $G^{*}_{21}$, which yields

[TABLE]

Now we estimate the integral $\int_{-1}^{1}R^{*}(t+\alpha u)K_{\zeta}(u)du$. Let $\theta_{0}$ be some constant. By the elementary formula

[TABLE]

we get

[TABLE]

with

[TABLE]

[TABLE]

By using

[TABLE]

we have

[TABLE]

which suggests

[TABLE]

Take $H=T$, $y=T^{\frac{1}{2}}$. Then clearly $y>1$. Thus we get

[TABLE]

by using $\sum_{n>1}\frac{d(n)}{n^{\frac{3}{4}}(\sqrt{n}-1)^{2}}\ll 1$. Noting that $H=T$, $t\asymp T^{\frac{1}{2}}$, by (4.2)-(4.4), we see

[TABLE]

Thus we complete the proof of Lemma 4.1 ∎

5. The mean value of $\Delta(q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2})$ in short intervals

In this section, we need the following Lemma.

Lemma 5.1**.**

$($* Hilbert’s inequality )$$( See e.g.[14] $)$    Let $x_{1}<x_{2}<\cdots<x_{n}$ be a sequence of real numbers. If there exists$~{}\delta>0$, such that $\min\limits_{s\neq r}|x_{r}-x_{s}|\geq\delta_{r}\geq\delta>0(1\leq r\leq n)$, then there exists an absolute constant $C$, such that*

[TABLE]

for arbitrary complex numbers $u_{1},u_{2},\cdots,u_{n}$.

Suppose $T\gg(q_{1}q_{2})^{\varepsilon}$ is a large parameter, $1\leq h_{0}\leq\frac{1}{2}\sqrt{T}$. Denote $\Delta^{*}(q_{1}q_{2}x)=\Delta(q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2})$. In this section we shall estimate the integral

[TABLE]

which would play an important role in the proof of Theorem 2.3. This type of integral was studied for the error term in the mean square of $\zeta(\frac{1}{2}+it)$ by Good [2], for the error term in the Dirichlet divisor problem by Jutila [10] and for the error term in Weyl’s law for Heisenberg manifold by Tsang and Zhai [16]. Here we follows the approach of Tsang and Zhai [16] and prove the following

Lemma 5.2**.**

The estimate

[TABLE]

holds uniformly for $1\leq h_{0}\leq\frac{1}{2}\sqrt{T}$.

Proof.

Write

[TABLE]

where

[TABLE]

From Corollary 2.1, we see that

[TABLE]

For $\int_{2}$, first we estimate the integral

[TABLE]

Let $T=2U$ in (3.1). Then

[TABLE]

Take $H=U$, y=\min\big{(}\frac{1}{2}Uh_{0}^{-1},U\log^{-6}U\big{)}. From [11, Lemma 4.1 and eq.(4.11)], we see

[TABLE]

Therefor

[TABLE]

We now estimate \int_{U}^{2U}\big{(}R_{0}(x+h_{0};y)-R_{0}(x;y)\big{)}^{2}dx. Set $\theta(h,l)=2\pi(\frac{hr_{2}}{q_{2}}+\frac{lr_{1}}{q_{1}})$. From (3.2), we have

[TABLE]

where

[TABLE]

From [11, Proof of Lemma 4.2], we get

[TABLE]

For the mean square of $F_{2}(x)$, we see

[TABLE]

where

[TABLE]

Write

[TABLE]

with

[TABLE]

[TABLE]

Let

[TABLE]

Using

[TABLE]

with $|d_{v}|<1$, we see

[TABLE]

Noting that $n_{1},n_{2}\leq y\leq\frac{1}{2}Uh_{0}^{-1}$, we have

[TABLE]

Then by the the first derivative test we get

[TABLE]

Noting $\sum_{n\leq N}d^{2}(n)\ll N\log^{3}N$, by using Lamma 5.1 and (5.8), we obtain

[TABLE]

By the elementary formulas

[TABLE]

we have

[TABLE]

where

[TABLE]

[TABLE]

It is easy to see that

[TABLE]

By using Taylor’s expansion, we have for $x\geq 100h_{0}^{2}$,

[TABLE]

which suggests

[TABLE]

in view of the fact $h_{0}^{2}<U$ and $n\leq y<U$. Hence,

[TABLE]

where we used the well-known estimate $\sum_{n\leq N}d^{2}(n)\ll N\log^{3}N$.

By the first derivative test, we have

[TABLE]

Using the integration by parts, we obtain

[TABLE]

which yields

[TABLE]

From (5.10)-(5.12), we get

[TABLE]

Combining (5.7), (5.9) and (5.13), we obtain

[TABLE]

which together with (5.5), (5.6) yields

[TABLE]

From (5.3), (5.4), and (5.14), it follows that

[TABLE]

which implies

[TABLE]

via a splitting argument. Then Lemma 5.2 follows from (5.1), (5.2), and (5.15). ∎

6. Proof of Theorem 2.3

In this section, we will give a proof of Theorem 2.3 by following the approach of [16]. We still write $\Delta^{*}(q_{1}q_{2}x)=\Delta(q_{1}q_{2}x;r_{1},q_{1},r_{2},q_{2})$. Define

[TABLE]

We need the following two lemmas.

Lemma 6.1**.**

[TABLE]

Proof.

From Corollary 2.1 with $k=2,4$, by Hölder’s inequality, we get

[TABLE]

which yields

[TABLE]

From (1.7), we see

[TABLE]

Thus, from the definition of $\Delta^{*}_{\pm}(q_{1}q_{2}t)$, we have

[TABLE]

Then by Cauchy-Schwarz’s inequality, we get

[TABLE]

which immediately implies Lemma 6.1. ∎

Lemma 6.2**.**

Suppose $2\leq H_{0}\leq\sqrt{T}$. Then

[TABLE]

Proof.

Since

[TABLE]

it is sufficient to prove that

[TABLE]

For $0<u_{1}<u_{2}\ll T$, it easy to see that

[TABLE]

Write $H_{0}=2^{\lambda}b$, such that $\lambda\in\mathbb{N}$ and $1\leq b<2$. Then for each $t\in[T,2T]$, we have

[TABLE]

Similar to the argument of the proof of Lemma 2 of [4], by using Lemma 5.2, we we can deduce that

[TABLE]

Thus we get Lemmma 6.2.∎

Now we finish the proof of Theorem 2.3. Let $P(t)=\Delta^{*}_{\pm}(q_{1}q_{2}t)$ and $Q(t)=\delta t^{\frac{1}{4}}$ for a sufficiently small $\delta>0$, and

[TABLE]

Then

[TABLE]

from Lemma 6.1 and Lemma 6.2, by taking $H_{0}=\delta T^{\frac{1}{2}}\mathcal{L}^{-7}$. For any point $t_{0}$, where $\omega(t_{0})>0$ and any $h\in[0,H_{0}]$, we see that $P(t_{0}+h)$ has the same sign as $P(t_{0})$, and $|P(t_{0}+h)|>\frac{1}{2}|Q(t_{0})|$.

Let

[TABLE]

From Corollary 2.1 and (6.2), using Cauchy-Schwarz’s inequality, we have

[TABLE]

which implies

[TABLE]

Thus the proof of Theorem 2.3 is completed. ∎

7. Proof of Theorem 2.4

Suppose $k\geq 3$ is a fixed odd integer and $T\gg(q_{1}q_{2})^{\varepsilon}$ is a large parameter. Set

[TABLE]

where $C_{k}$ is defined in (2.1).

By Theorem 2.3, there exists $t\in[T,2T]$ such that $\delta\Delta(q_{1}q_{2}u;r_{1},q_{1},r_{2},q_{2})>c_{5}t^{\frac{1}{4}}$ for any $u\in[t,t+H_{0}]$, with $H_{0}=c_{4}\sqrt{T}\mathcal{L}^{-7}$. Thus

[TABLE]

which yields

[TABLE]

with

[TABLE]

Thus we get

[TABLE]

which immediatly implies Theorem 2.4. ∎