Sum of the triple divisor function of mixed powers

Li Zhou; Liqun Hu; Li Zhou; Liqun Hu

doi:10.3934/math.2022713

AIMS Mathematics

2022, Volume 7, Issue 7: 12885-12896. doi: 10.3934/math.2022713

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Research article

Sum of the triple divisor function of mixed powers

Li Zhou ¹,
Liqun Hu ^{1,2
,
,}

1.
Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China
2.
Institute of Mathematics and Interdisciplinary Sciences, Nanchang University, Nanchang 330031, Jiangxi, China

Received: 05 January 2022 Revised: 17 April 2022 Accepted: 27 April 2022 Published: 05 May 2022
MSC : 11P32, 11P05, 11P55

Let $d_3(n)$ denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum

$\sum\limits_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d_3(n_1^2+n_2^2+n_3^k)$

with $n_1, n_2, n_3\in \mathbb{Z}^+$ and $k \geqslant 3$ be an integer. Previously only the case of $k=2$ is studied.

Keywords:

Citation: Li Zhou, Liqun Hu. Sum of the triple divisor function of mixed powers[J]. AIMS Mathematics, 2022, 7(7): 12885-12896. doi: 10.3934/math.2022713

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Abstract

Let $d_3(n)$ denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum

$\sum\limits_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d_3(n_1^2+n_2^2+n_3^k)$

with $n_1, n_2, n_3\in \mathbb{Z}^+$ and $k \geqslant 3$ be an integer. Previously only the case of $k=2$ is studied.

1. Introduction

Let $X>1$ be a large integer and $d_t(n)$ denote the t-th divisor function. For $t=2$ , we denote $d(n):=d_2(n)$ . In 2012, Guo and Zhai ^[3] considered the asymptotic formula of the sum

$\begin{eqnarray*} \sum\limits_{ 1 \leqslant n_1,n_2,n_3 \leqslant X^{\frac{1}{2}}}d(n_1^2+n_2^2+n_3^2)=c_1X^{\frac{3}{2}}({\log X})^2+c_2X^{\frac{3}{2}}+O(X^{\frac{4}{3}+\epsilon}), \end{eqnarray*}$

where $c_1$ and $c_2$ are constants. The error term was refined to $O(X\log^7 X)$ by Zhao ^[10]. Later, Hu ^[4] considered the case of $n_1^2+n_2^2+n_3^2+n_4^2$ . In 2016, Lü and Mu ^[7] considered the asymptotic formula of the sum

$\begin{eqnarray*} \sum\limits_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d(n_1^2+n_2^2+n_3^k)=c_3X^{1+\frac{1}{k}}{\log X}+c_4X^{1+\frac{1}{k}}+O(X^{1+\frac{1}{k}-\delta(k)+\epsilon}), \end{eqnarray*}$

with $k \geqslant 3$ . Here $c_3$ , $c_4$ are constants and

$\delta(k)=\left\{ \begin{aligned} &\frac{5}{42}, &k&=3 , \\ &\frac{1}{16}, &k&=4 , \\ &\frac{1}{40}, &k&=5 , \\ &\frac{1}{k2^{k-1}}, &6& \leqslant k \leqslant7 , \\ &\frac{1}{2k^2(k-1)}, &k& \geqslant 8. \end{aligned} \right.$

For $t=3$ , recently, Sun and Zhang ^[8] considered the asymptotic formula of the sum

$\begin{eqnarray*} \sum_{ 1 \leqslant n_1,n_2,n_3 \leqslant X^{\frac{1}{2}}}d_3(n_1^2+n_2^2+n_3^2)=c_5X^{\frac{3}{2}}\log^2 X+c_6X ^{\frac{3}{2}}{\log X}+c_7X^{\frac{3}{2}}+O(X^{\frac{3}{2}-{\frac{1}{8}}+\epsilon}). \end{eqnarray*}$

where $c_5$ , $c_6$ and $c_7$ are constants. Hu and Yang ^[5] also considered the case of $n_1^2+n_2^2+n_3^2+n_4^2$ .

In this paper, inspired by Lü and Mu ^[7], we want to consider the asymptotic formula of the sum

$\begin{eqnarray*} \sum_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d_3(n_1^2+n_2^2+n_3^k) \end{eqnarray*}$

with $k \geqslant 3$ . In order to state our result, let $G_k(a,b,q)$ be the Gauss sum

$G_k(a,b,q)=\sum\limits_{x \bmod q}e \left(\frac{ax^k+bx}{q} \right).$

and denote $G(a,b,q):=G_2(a,b,q)$ . For $0 \leqslant j \leqslant 2$ , define

$\begin{eqnarray*} A_j(q)=\sum_{b=1}^q e\left(-\frac{ab}{q}\right)c_{j+1}(b,q), \end{eqnarray*}$

where $a > 0$ is an integer. The coefficients $c_j(b,q)$ are sums of terms of the form

$\begin{eqnarray*} \sum_{{b_1b_2\equiv b}(\bmod q)}f(b_1) \end{eqnarray*}$

for some function $f$ . The number of terms in $c_j(b,q)$ depends only on $k$ . More precisely, The coefficients $c_j(b,q)$ are given explicitly in ^[1, (2.13)]. From ^[2, (4.8)] we can get

$\begin{eqnarray} A_j(q) \ll _k q^{-1}. \end{eqnarray}$

(1.1)

For $0 \leqslant i \leqslant j \leqslant 2$ , define

$\begin{eqnarray} \mathfrak{S}=\frac{1}{j!}\sum_{q=1}^{\infty}{{\sum_{a \bmod q}}^*}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3} \end{eqnarray}$

(1.2)

and

$\begin{eqnarray} \mathfrak{I}=\int_{-\infty}^{\infty}\left(\int_{0}^{3}e(-\beta u )({\log u})^i \,du\right)\left(\int_{0}^1e(\beta v^2)\,dv\right)^2\left(\int_{0}^1e(\beta v^k)dv\right)d{\beta}. \end{eqnarray}$

(1.3)

Our result is the following theorem.

Theorem 1.1. let $X>1$ and $k \geqslant 3$ be integers and let $d_3(n)$ denote the 3-th divisor function. We have

$\begin{eqnarray*} &&\sum_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d_3(n_1^2+n_2^2+n_3^k)\\ &=&\sum_{j=0}^{2}\mathfrak{S} \sum_{i=0}^j C_j^i \mathfrak{I} X^{1+\frac{1}{k}}({\log X})^{j-i} +O(X^{1+{\frac{1}{k}-\delta(k)+\epsilon}}), \end{eqnarray*}$

where $\mathfrak{S}$ are defined in (1.2), $\mathfrak{I}$ are defined in (1.3), and

$\delta(k)=\left\{ \begin{aligned} &\frac{1}{15}, &&k=3,\\ &\frac{1}{k2^{k-1}}, &&4 \leqslant k \leqslant 7,\\ &\frac{1}{2k^2(k-1)}, &&k \geqslant 8.\\ \end{aligned} \right.$

To prove our theorem, we use the circle method and some strategies in the work of ^[7].

2. Outline of the method

Let $X>1$ and $k \geqslant 3$ be integers. In order to apply the circle method, we choose the parameters $P$ and $Q$ such that

$\begin{eqnarray*} P=X^{{\theta}} \quad {\rm{and}} \quad Q=X^{1-{\theta}}, \end{eqnarray*}$

where

$\theta=\left\{ \begin{aligned} &\frac{k+1}{6k+2}, &&3 \leqslant k \leqslant 9,\\ &\frac{2}{k+2}, &&k \geqslant 10.\\ \end{aligned} \right.$

Obviously, we have $Q> X^{1/2}$ . By Dirichlet's lemma on rational approximations, each $\alpha\in [Q^{-1},1+Q^{-1}]$ can be written in the form

$\begin{eqnarray} \alpha=\frac{a}{q}+\beta,\quad |\beta| \leqslant \frac{1}{qQ} \end{eqnarray}$

(2.1)

for some integers $a$ , $q$ with $1 \leqslant a \leqslant q \leqslant Q$ and $(a,q)=1$ . We denote by $\mathfrak{M}(q,a)$ the set of $\alpha$ satisfying (2.1) and define the major arcs and the minor arcs as follows.

$\begin{eqnarray*} \mathfrak{M}=\bigcup_{q \leqslant P} \bigcup_{\substack{1 \leqslant a \leqslant q \\ (a,q)=1}}\mathfrak{M}(q,a), \quad \mathfrak{m}=\bigg[\frac{1}{Q},1+\frac{1}{Q}\bigg]\setminus\mathfrak{M}. \end{eqnarray*}$

Let

$\begin{eqnarray} &&F(\alpha,X)=\sum_{1 \leqslant n \leqslant 3X}{d_3(n)e(-n\alpha)},\\ &&S_k(\alpha,X)=\sum_{1 \leqslant m \leqslant X^{\frac{1}{k}}}{e(m^k\alpha)}. \end{eqnarray}$

(2.2)

Our Theorem 1.1 is a consequence of the following Proposition 2.1. We will give a proof of Proposition 2.1 in Section 3.

Proposition 2.1. Let $F(\alpha,X)$ and $S_k(\alpha,X)$ be defined as in (2.2). For $\alpha \in \mathfrak{M}$ , we have

$\begin{eqnarray*} &&\int\limits_{\mathfrak{M}}S_2^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d\alpha\\ &=&\sum_{j=0}^{2}\mathfrak{S}\sum_{i=0}^j C_j^i \mathfrak{I} X^{1+\frac{1}{k}}({\log X})^{j-i}+O(X^{1+{\frac{1}{k}-\delta'(k)+\epsilon}}), \end{eqnarray*}$

where $\mathfrak{S}$ and $\mathfrak{I}$ are defined as in Theorem 1.1 and

$\delta'(k)=\left\{ \begin{aligned} &\frac{k+1}{2k(3k+1)}, &&3 \leqslant k \leqslant 9,\\ &\frac{2}{k(k+2)}, &&k \geqslant 10.\\ \end{aligned} \right.$

Now we can get Theorem 1.1 from Proposition 2.1.

Proof of Theorem 1.1. Applying the circle method, we get

$\begin{eqnarray*} &&\sum_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d_3(n_1^2+n_2^2+n_3^k)\\ &=&\int\limits_{\frac{1}{Q}}^{1+\frac{1}{Q}}S^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}\\ &=&\int\limits_{\mathfrak{M}}S^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}+\int\limits_{\mathfrak{m}}S^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d\alpha. \end{eqnarray*}$

To estimate the contribution from the minor arcs, we note that each $\alpha\in \mathfrak{m}$ can be written as (2.1) for $P<q \leqslant Q$ and $1 \leqslant a \leqslant q$ with $(q,a)=1$ . In order to make the error term in the asymptotic formula as small as possible, we consider it in two cases.

In the case of $3 \leqslant k \leqslant 7$ , for $\alpha\in \mathfrak{m}$ , by Weyl's inequality we have

$\begin{eqnarray*} S_k(\alpha,X)\ll X^{\frac{1}{k}+\epsilon}(P^{-1}+X^{-\frac{1}{k}}+QX^{-1})^{\frac{1}{2^{k-1}}}. \end{eqnarray*}$

Combining Cauchy's inequality and Hua's inequality, we have

$\begin{eqnarray} &&\int\limits_{\mathfrak{m}}S^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}\\ &\ll& \max\limits_{\alpha\in \mathfrak{m}}|S_k(\alpha,X)| \left(\int_{0}^{1}|S_2(\alpha,X)|^4\,d\alpha\right)^{\frac{1}{2}}\left(\int_{0}^{1}|F(\alpha,X)|^2\,d\alpha\right)^{\frac{1}{2}}\\ &\ll& X^{1+\frac{1}{k}+\epsilon}P^{-2^{(1-k)}}+X^{1+\frac{1}{k}(1-2^{(1-k)})+\epsilon}+ X^{1+\frac{1}{k}-2^{(1-k)}+\epsilon}Q^{2^{(1-k)}}. \end{eqnarray}$

(2.3)

In the case of $k \geqslant 8$ , for $\alpha\in \mathfrak{m}$ , we take (^[7, Lemma 1.6]) in place of Weyl's inequality and get

$\begin{eqnarray*} S_k(\alpha,X)\ll X^{\frac{1}{k}+\epsilon}(P^{-1}+X^{-\frac{1}{k}}+QX^{-1})^{{\frac{1}{{2k}{(k-1)}}}}. \end{eqnarray*}$

Similarly with $(2.3)$ , we have

$\begin{eqnarray} &&\int\limits_{\mathfrak{m}}S^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d\alpha\\ &\ll& X^{1+\frac{1}{k}+\epsilon}P^{-\frac{1}{{2k}^({k-1})}}+X^{1+\frac{1}{k}(1-\frac{1}{1-2k({k-1})})+\epsilon} +X^{1+\frac{1}{k}-\frac{1}{{2k}({k-1})}+\epsilon}Q^{\frac{1}{{2k}{(k-1)}}}. \end{eqnarray}$

(2.4)

From Propositions 2.1, (2.3) and (2.4), Theorem 1.1 follows by taking $Q=X^{1-{\theta}}$ with

$\theta=\left\{ \begin{aligned} &\frac{k+1}{6k+2}, &&3 \leqslant k \leqslant 9,\\ &\frac{2}{k+2}, &&k \geqslant 10.\\ \end{aligned} \right.$

3. Proof of Proposition 2.1

For $\alpha\in \mathfrak{M}$ , we write $\alpha=a/q+\beta$ , $1 \leqslant a \leqslant q \leqslant P$ and $|\beta| \leqslant 1/(qQ)$ with $(a,q)=1$ . Then we have

$\begin{eqnarray} &&\int\limits_{\mathfrak{M}}S_2^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}\\ &=&\sum_{q \leqslant P} \sideset{}{^*}\sum_{a \bmod q}\int\limits_{\mathfrak{M}(q,a)}S_2^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}\\ &=&\sum_{q \leqslant P} \int\limits_{|\beta| \leqslant \frac{1}{qQ}}\sideset{}{^*}\sum_{{a \bmod q}}S_2^2\left(\frac{a}{q}+\beta,X\right)S_k\left(\frac{a}{q}+\beta,X\right)F\left(\frac{a}{q}+\beta,X\right)d{\beta}\\. \end{eqnarray}$

(3.1)

First we need to estimate $F(a/q+\beta,X)$ .

Lemma 3.1. Suppose that $(a,q)=1$ , $q \leqslant P \leqslant x^{1/3}$ and $|\beta| \leqslant 1/(qQ)$ . We have

$\begin{eqnarray*} F\left(\frac{a}{q}+\beta,X\right)=\sum_{j=0}^{2}{A_j(q)I_{j}(\beta)}+O(P^{3+\epsilon}+X^{\eta+\epsilon}P), \end{eqnarray*}$

where $\eta=2/5$ , $A_j(q)$ are defined in (1.2) and

$\begin{eqnarray} I_j(\beta)=\int_{1}^{3X}e(-\beta u)\frac{\log^j u}{j!}\,du. \end{eqnarray}$

(3.2)

Proof. This is lemma 3.2 in ^[2] for $k=3$ and $l=3$ . Integrating by parts, we have

$\begin{eqnarray} I_{j}(\beta) \ll _k X^{\epsilon} \min \left\{X ,|\beta|^{-1}\right\}. \end{eqnarray}$

(3.3)

Next we need to estimate $S_2(a/q+\beta,X)$ and $S_k(a/q+\beta,X)$ .

Lemma 3.2. Suppose that $(a,q)=1$ , $q \leqslant P \leqslant X^{\frac{1}{3}}$ and $|\beta| \leqslant 1/(qQ)$ . We have

$\begin{eqnarray*} &&S_2\left(\frac{a}{q}+\beta,X\right)=\frac{G(a,0,q)}{q}\Psi_0(\beta)+\sum_{-3q/2 < b < 3q/2}{G(a,b,q)}\Psi(b,q,\beta),\\ &&S_k\left(\frac{a}{q}+\beta,X\right)=\frac{G_k(a,0,q)}{q}\Psi_k(\beta)+O\left(q^{\frac{1}{2}+\epsilon}(1+X|\beta|)^{\frac{1}{2}}\right) \end{eqnarray*}$

where $G_k(a,b,q)$ are defined as in (1.1), $\Psi_0(\beta)$ and $\Psi_k(\beta)$ are the integral

$\begin{eqnarray*} \Psi_0(\beta)=\int_{0}^{X^{\frac{1}{2}}}e(\beta u^2)\,du,\\ \Psi_k(\beta)=\int_{0}^{X^{\frac{1}{k}}}e(\beta u^k)\,du \end{eqnarray*}$

and $\Psi(b,q,\beta)$ satisfies

$\begin{eqnarray} \sum_{-3q/2 < b < 3q/2}|\Psi(b,q,\beta|\ll {\log {(q+2)}}. \end{eqnarray}$

(3.4)

Proof. The first formula is come from ^[10, Lemma 4.1]. We can also find the second formula in ^[9, Theorem 4.1].

From Lemmas 3.1 and 3.2, we have

$\begin{eqnarray} &&\sideset{}{^*}\sum_{a\bmod q} S^2\left(\frac{a}{q}+\beta,X\right)S_k\left(\frac{a}{q}+\beta,X\right)F\left(\frac{a}{q}+\beta,X\right)\\ &=&\sum_{i=1}^{3}T_i(q,\beta)S_k\left(\frac{a}{q}+\beta,X\right)\\ &&+O\bigg((P^{3+\epsilon}+X^{\eta+\epsilon})\sideset{}{^*}\sum_{a\bmod q}\bigg|S\left(\frac{a}{q}+\beta,X\right)\bigg|^2\bigg|S_k\left(\frac{a}{q}+\beta,X\right)\bigg|\bigg), \end{eqnarray}$

(3.5)

where

$\begin{eqnarray*} T_i(q,\beta)&=&\sum_{j=0}^{2} \sideset{}{^*}\sum_{a\bmod q}A_j(q)I_j(\beta)C_2^{i-1}\left(\frac{{G(a,0,q)}\Psi_0(\beta)}{q}\right)^{3-i}\\ &&\times\left(\sum_{-3q/2 < b < 3q/2}{G(a,b,q)}\Psi(b,q,\beta)\right)^{i-1}. \end{eqnarray*}$

On inserting (3.5) into (3.1), we have

$\begin{eqnarray} &&\int\limits_{\mathfrak{M}}S_2^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}\\ &=&\sum_{i=1}^{3}\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_i(q,\beta)S_k\left(\frac{a}{q}+\beta,X\right)d{\beta}\\ &&+O\left((P^{3+\epsilon}+X^{\eta+\epsilon})\int\limits_{\mathfrak{M}}|S_2(\alpha,X)|^2|S_k(\alpha,X)|d{\alpha}\right). \end{eqnarray}$

(3.6)

It follows from integration by parts together with trivial bounds that

$\begin{eqnarray} \Psi_0(\beta) \ll \left(\frac{X}{1+|\beta|X}\right)^{\frac{1}{2}},\quad \Psi_k(\beta) \ll \left(\frac{X}{1+|\beta|X}\right)^{\frac{1}{k}}. \end{eqnarray}$

(3.7)

3.1. Estimate of $T_1(q,\beta)S_k(a/q+\beta,X)$

By the definition of $T_1(q,\beta)$ and Lemma 3.2, we have

$\begin{eqnarray*} &&\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_i(q,\beta)S_k\left(\frac{a}{q}+\beta,X\right)d{\beta}\\ &=&\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_1(q,\beta){\frac{G_k(a,0,q)}{q}\Psi_k(\beta)}d{\beta}\\ &&+\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_i(q,\beta)O\left(q^{\frac{1}{2}+\epsilon}(1+X|\beta|)^{\frac{1}{2}}\right)d{\beta}\\ &=&\sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3} \int\limits_{|\beta| \leqslant \frac{1}{qQ} }I_j(\beta)\Psi_0^2(\beta)\Psi_k(\beta)d{\beta}\\ &&+\sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)q^{\frac{1}{2}+\epsilon}}{q^2} \int\limits_{|\beta| \leqslant \frac{1}{qQ} }I_j(\beta)\Psi_0^2(\beta)(1+X|\beta|)^{\frac{1}{2}}d{\beta}\\ &:=&\sum_1+\sum_2. \end{eqnarray*}$

By Lemma 3.4 in ^[6] and Theorem 4.2 in ^[9], we have

$\begin{eqnarray} G(a,b,q)\ll q^{\frac{1}{2}}, \quad G_k(a,0,q)\ll q^{1-\frac{1}{k}}. \end{eqnarray}$

(3.8)

To $\sum_1$ , we extend the integration over $|\beta| \leqslant 1/(qQ)$ to $(-\infty,\infty)$ and the error term is

$\begin{eqnarray*} &&\sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3} \int\limits_{|\beta| > \frac{1}{qQ} }I_j(\beta)\Psi_0^2(\beta)\Psi_k(\beta)d{\beta}\\ &\ll& \sum_{q \leqslant P}{q^{-1-\frac{1}{k}}}\int\limits_{|\beta| > \frac{1}{qQ} }|\beta|^{-2-{\frac{1}{k}}}d{\beta}\\ &\ll& \sum_{q \leqslant P}{q^{-1-\frac{1}{k}}} (qQ)^{1+{\frac{1}{k}}}\\ &\ll& XQ^{\frac{1}{k}}. \end{eqnarray*}$

Here we use the upper bounds (1.1), (3.3), (3.7) and (3.8). Thus we have

$\begin{eqnarray*} &&\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_1(q,\beta){\frac{G_k(a,0,q)}{q}\Psi_k(\beta)}d{\beta}\\ &=&\sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3} \int_{-\infty}^{\infty}I_j(\beta)\Psi_0^2(\beta)\Psi_k(\beta)d{\beta}\\ &&+O(XQ^{\frac{1}{k}}). \end{eqnarray*}$

Next, we estimate the integral on the right hand side. From the definitions of $I_j(\beta)$ , $\Psi_0(\beta)$ and $\Psi_k(\beta)$ , we obtain

$\begin{eqnarray*} &&\int_{-\infty}^{\infty}I_j(\beta)\Psi_0^2(\beta)\Psi_k(\beta)d{\beta}\\ &=&\int_{-\infty}^{\infty}\left(\int_{1}^{3X}e(-\beta u )\frac{\log^j u }{j!}du\right)\left(\int_{0}^{X^{\frac{1}{2}}}e(\beta v^2)dv\right)^2\left(\int_{0}^{X^{\frac{1}{k}}}e(\beta v^k)dv\right)\,d{\beta}\\ &=&\frac{X^{1+\frac{1}{k}}}{j!}\int_{-\infty}^{\infty}\left(\int_{1}^{3X}e(-\beta u ){\log^j u} \,du\right)\left(\int_{0}^1e(\beta X v^2)\,dv\right)^2\left(\int_{0}^1e(\beta X v^k)\,dv\right)\,d{\beta}\\ &=&\frac{X^{1+\frac{1}{k}}}{j!}\int_{-\infty}^{\infty}\left(\int_{\frac{1}{X}}^{3}e(-\beta u )({\log (Xu)})^j du\right)\left(\int_{0}^1e(\beta v^2)dv\right)^2\left(\int_{0}^1e(\beta v^k)\,dv\right)\,d{\beta}\\ &=&\frac{X^{1+\frac{1}{k}}}{j!}\int_{-\infty}^{\infty}\left(\int_{0}^{3}e(-\beta u )({\log (Xu)})^j du\right)\left(\int_{0}^1e(\beta v^2)dv\right)^2\left(\int_{0}^1e(\beta v^k)\,dv\right)\,d{\beta}\\ &&+O({X^{\frac{1}{k}+\epsilon}}). \end{eqnarray*}$

Here

$\begin{eqnarray*} &&\frac{X^{1+\frac{1}{k}}}{j!}\int_{-\infty}^{\infty}\left(\int_{0}^{\frac{1}{X}}e(-\beta u )(\log (Xu))^jdu\right)\left(\int_{0}^1e(\beta v^2)dv\right)^2\left(\int_{0}^1e(\beta v^k)dv\right)d{\beta}\\ &\ll& X^{\frac{1}{k}+\epsilon}. \end{eqnarray*}$

By $(1.1)$ and $(3.8)$ , we have

$\begin{eqnarray*} \sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3}\ll 1. \end{eqnarray*}$

Then we split $\log (Xu)$ and obtain

$\begin{eqnarray*} &&\sum_{q \leqslant P}\int_{|\beta| \leqslant \frac{1}{qQ} }T_1(q,\beta){\frac{G_k(a,0,q)}{q}\Psi_k(\beta)}\,d{\beta}\\ &=&\sum_{j=0}^{2}\frac{1}{j!}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3}\sum_{i=0}^jC_j^i\mathfrak{I} X^{1+\frac{1}{k}}(\log X)^{j-i}\\ &&+O(XQ^{\frac{1}{k}}+X^{\frac{1}{k}+\epsilon}), \end{eqnarray*}$

where $\mathfrak{I}$ are defined as in $(1.3)$ . Further, we extend the summation over $q \leqslant P$ to all positive integers and the estimate of the error term is

$\begin{eqnarray*} &&\sum_{j=0}^{2}\frac{1}{j!}\sum_{q > P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^2(a,0,q)G_k(a,0,q)}{q^3}\sum_{i=0}^jC_j^i\mathfrak{I} X^{1+\frac{1}{k}}(\log X)^{j-i}\\ &\ll& XQ^{\frac{1}{k}}. \end{eqnarray*}$

Thus we have

$\begin{eqnarray*} \sum_1 &=& \sum_{j=0}^{2}\sum_{q \leqslant P}\int_{|\beta| \leqslant \frac{1}{qQ} }T_1(q,\beta){\frac{G_k(a,0,q)}{q}\Psi_k(\beta)}d{\beta}\\ &=&\sum_{j=0}^{2}\mathfrak{S} \sum_{i=0}^jC_j^i\mathfrak{I} X^{1+\frac{1}{k}}(\log X)^{j-i} +O(XQ^{\frac{1}{k}}+X^{\frac{1}{k}+\epsilon}), \end{eqnarray*}$

where $\mathfrak{S}$ and $\mathfrak{I}$ defined in (1.2) and (1.3).

To $\sum_2$ , by (1.1), (3.3), (3.7) and (3.8) we have

$\begin{eqnarray*} \sum_2 \ll\sum_{q \leqslant P}{q^{-\frac{1}{2}+\epsilon}}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }|\beta|^{-1}X(1+X|\beta|)^{-\frac{1}{2}}d{\beta}\ll X^{\frac{3}{2}+\epsilon}Q^{-\frac{1}{2}}. \end{eqnarray*}$

To sum up, we have

$\begin{eqnarray*} &&\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_i(q,\beta)S_k\left(\frac{a}{q}+\beta,X\right)d{\beta}\\ &=&\sum_{j=0}^{2}\mathfrak{S} \sum_{i=0}^jC_j^i\mathfrak{I} X^{1+\frac{1}{k}}(\log X)^{j-i} +O(XQ^{\frac{1}{k}}+X^{\frac{3}{2}+\epsilon}Q^{-\frac{1}{2}}+X^{\frac{1}{k}+\epsilon}), \end{eqnarray*}$

where $\mathfrak{S}$ and $\mathfrak{I}$ are defined in (1.2) and (1.3).

3.2. Estimate of $T_i(q,\beta)S_k(a/q+\beta,X)$ with $i=2,3$

Now by the definition of $T_i(q,\beta)$ and Lemma 3.2, we have

$\begin{eqnarray*} &&\sum_{q \leqslant P}\int\limits_{|\beta| \leqslant \frac{1}{qQ} }T_i(q,\beta)S_k(a/q+\beta,X)d\beta\\ &=&C_2^{i-1}\sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^{3-i}(a,0,q)G_k(a,0,q)}{q^{4-i}}\\ &&\times\int\limits_{|\beta| \leqslant \frac{1}{qQ} }I_j(\beta)\Psi_0^{3-i}(\beta)\Psi_k(\beta)\left(\sum_{-3q/2 < b < 3q/2}{G(a,b,q)}\Psi(b,q,\beta)\right)^{i-1}d{\beta}\\ &&+C_2^{i-1}\sum_{j=0}^{2}\sum_{q \leqslant P}\sideset{}{^*}\sum_{a\bmod q}\frac{A_j(q)G^{3-i}(a,0,q)}{q^{3-i}} \int\limits_{|\beta| \leqslant \frac{1}{qQ}}I_j(\beta)\Psi_0^{3-i}(\beta) \\&&\times \left(\sum_{-3q/2 < b < 3q/2}{G(a,b,q)}\Psi(b,q,\beta)\right)^{i-1} O\left(q^{\frac{1}{2}+\epsilon}(1+X|\beta|)^{\frac{1}{2}}\right)d{\beta}\\ &:=&\sum_3+\sum_4.\\ \end{eqnarray*}$

Combining Lemma 3.3 with $(1.1)$ , $(3.3)$ , $(3.4)$ , $(3.7)$ and $(3.8)$ , for $i=2,3$ we have

$\begin{eqnarray*} \sum_3\ll \sum_{q \leqslant P}q^{\frac{3-i}{2}}q^{i-4}q^{1-\frac{1}{k}}q^{\frac{i-1}{2}} X^{\frac{3}{2}+\frac{1}{k}-\frac{i}{2}} \ll X^{\frac{i}{2}+\frac{1}{2}}Q^{1+\frac{1}{k}-i}. \end{eqnarray*}$

For $i=2$ ,

$\begin{eqnarray*} \sum_4\ll \sum_{q \leqslant P}q^{\frac{1}{2}}q^{-1}q^{\frac{1}{2}+\epsilon}q^{\frac{1}{2}} X^{\frac{1}{2}+\epsilon}\ll X^{2+\epsilon}Q^{-\frac{3}{2}-\epsilon}. \end{eqnarray*}$

For $i=3$ ,

$\begin{eqnarray*} \sum_4\ll \sum_{q \leqslant P}q^{\frac{1}{2}+\epsilon}q X^{\frac{1}{2}}(qQ)^{-\frac{1}{2}}\ll X^{\frac{5}{2}+\epsilon}Q^{-\frac{5}{2}-\epsilon}. \end{eqnarray*}$

3.3. Estimate of the O-term

By Hua's inequality, the contribution of O-term is bounded by

$\begin{eqnarray*} &&(P^{3+\epsilon}+X^{\eta+\epsilon}P)\int\limits_\mathfrak{M}|S_2{(\alpha,X)}|^2|S_k({\alpha,X})|\,d\alpha \\ &\ll&(P^{3+\epsilon}+X^{\eta+\epsilon}P)\left(\int\limits_\mathfrak{M}|S_2{(\alpha,X)}|^4\,d\alpha\right)^{\frac{1}{2}}\left(\int\limits_\mathfrak{M}|S_k({\alpha,X})|^2\,d\alpha\right)^{\frac{1}{2}}\\ &\ll&X^{\frac{1}{2}+\frac{1}{2k}}P^{3+\epsilon}+X^{\frac{1}{2}+\frac{1}{2k}+\eta+\epsilon}P\\ &=&X^{\frac{7}{2}+\frac{1}{2k}+\epsilon}Q^{-3-\epsilon}+X^{\frac{3}{2}+\frac{1}{2k}+\eta+\epsilon}Q^{-1}. \end{eqnarray*}$

3.4. Completion of the proof

From above, we have

$\begin{eqnarray*} &&\int\limits_{\mathfrak{M}}S_2^2(\alpha,X)S_k(\alpha,X)F(\alpha,X)d{\alpha}\nonumber\\ &=&\sum_{j=0}^{2}\mathfrak{S} \sum_{i=0}^jC_j^i\mathfrak{I} X^{1+\frac{1}{k}}(\log X)^{j-i} +O(XQ^{\frac{1}{k}}+X^{\frac{3}{2}+\epsilon}Q^{-\frac{1}{2}}+X^{\frac{1}{k}+\epsilon}\\ &&+X^{\frac{3}{2}}Q^{-1+\frac{1}{k}}+X^2Q^{-2+\frac{1}{k}}+X^{2+\epsilon}Q^{-\frac{3}{2}-\epsilon}+X^{\frac{5}{2}+\epsilon}Q^{-\frac{5}{2}-\epsilon}\\ &&+X^{\frac{7}{2}+\frac{1}{2k}+\epsilon}Q^{-3-\epsilon}+X^{\frac{3}{2}+\frac{1}{2k}+\eta+\epsilon}Q^{-1}). \end{eqnarray*}$

By taking $Q=X^{1-{\theta}},$ where

$\theta=\left\{ \begin{aligned} &\frac{k+1}{6k+2}, &&3 \leqslant k \leqslant 9,\\ &\frac{2}{k+2}, &&k \geqslant 10.\\ \end{aligned} \right.$

Comparing each item in the error term, we have for $3 \leqslant k \leqslant 9$ ,

$\begin{eqnarray*} &&\int\limits_\mathfrak{M} F(\alpha,X)S_2^2(\alpha,X)S_k(\alpha,X)d{\alpha}\\ &=&\sum_{j=0}^{2}\mathfrak{S} \sum_{i=0}^jC_j^i\mathfrak{I} X^{1+\frac{1}{k}}(\log X)^{j-i}+O\left(X^{1+\frac{1}{k}-\frac{k+1}{2k(3k+1)}+\epsilon}\right). \end{eqnarray*}$

For $k \geqslant 10$ ,

Then we finish the proof of Proposition 2.1.

Acknowledgments

This work is supported by Natural Science Foundation of China (Grant Nos. 11761048), Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars (Grant Nos. 20212ACB211007), Social Science Planning Project of Jiangxi Province of 2020 Annual (Grant Nos. 20JY06), Tender subject for key research base of Humanities and Social Sciences in Colleges and universities in Jiangxi Province (Grant Nos. JD20109), Culture and Art Science Planning Project of Jiangxi Province of 2020 Annual (Grant Nos. YG2020129) and Basic education research project of Jiangxi Province (Grant Nos. SZUNDSX2019-984). The authors would like to express their thanks to the referee for many useful suggestions and comments on the manuscript.

Conflict of interest

The authors declare there is no conflict of interests.

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[3]	R. T. Guo, W. G. Zhai, Some problems about the ternary quadratic form $m_1^2 +m_2^2 +m_3^2$ , Acta Arith., bf 156 (2012), 101–121.
[4]	L. Q. Hu, An asymptotic formula related to the divisors of the quaternary quadratic form, Acta Arith., 166 (2014), 129–140. https://doi.org/10.4064/aa166-2-2 doi: 10.4064/aa166-2-2
[5]	L. Q. Hu and L. Yang, Sums of the triple divisor function over values of a quaternary quadratic form, Acta Arith., 183 (2018), 63–85. https://doi.org/10.4064/aa170120-20-10 doi: 10.4064/aa170120-20-10
[6]	L. K. Hua, Introduction to Number Theory, Science Press, Beijing, 1957 (in Chinese).
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[9]	R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Math., vol. 125, Cambridge University, Cambridge, 1997.
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AIMS Mathematics

Sum of the triple divisor function of mixed powers

Related Papers:

Abstract

1. Introduction

2. Outline of the method

3. Proof of Proposition 2.1

3.1. Estimate of $T_1(q,\beta)S_k(a/q+\beta,X)$

3.2. Estimate of $T_i(q,\beta)S_k(a/q+\beta,X)$ with $i=2,3$

3.3. Estimate of the O-term

3.4. Completion of the proof

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

Sum of the triple divisor function of mixed powers

Related Papers:

Abstract

1. Introduction

2. Outline of the method

3. Proof of Proposition 2.1

3.1. Estimate of T1(q,β)Sk(a/q+β,X) T_1(q,\beta)S_k(a/q+\beta,X)

3.2. Estimate of Ti(q,β)Sk(a/q+β,X) T_i(q,\beta)S_k(a/q+\beta,X) with i=2,3 i=2,3

3.3. Estimate of the O-term

3.4. Completion of the proof

Acknowledgments

Conflict of interest

References

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3.1. Estimate of $T_1(q,\beta)S_k(a/q+\beta,X)$

3.2. Estimate of $T_i(q,\beta)S_k(a/q+\beta,X)$ with $i=2,3$