The sum of multidimensional divisor function over values of quadratic polynomial

TL;DR

This paper derives explicit asymptotic formulas for the sum of the multidimensional divisor function over values of a positive quadratic polynomial in three or more variables, using the circle method.

Contribution

It provides the first explicit asymptotic formulas for sums involving the multidimensional divisor function over quadratic polynomial values in multiple variables.

Findings

Asymptotic formulas are obtained for the sum T_{k,F}(X).

Results apply to quadratic polynomials with non-zero discriminant.

The circle method is effectively used to analyze these sums.

Abstract

Let $F({\bf x})={\bf x}^tQ_m{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge 1}^{\ell}$, $Q_m\in {\rm M}_{\ell}(\mathbb{Z})$ is an $\ell\times\ell$ matrix and its discriminant $\det\left(Q_m^t+Q_m\right)\neq 0$. It gives explicit asymptotic formulas for the following sum \[ T_{k,F}(X)=\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}\tau_{k}\left(F({\bf x})\right) \] with the help of the circle method. Here $\tau_{k}(n)=\#\{(x_1,x_2,...,x_{k})\in\mathbb{N}^{k}: n=x_1x_2...x_{k}\}$ with $k\in\mathbb{Z}_{\ge 2}$ is the multidimensional divisor function.

Equations

$$T_{k,F}(X)=\sum _{\mathbf{x}\in [1,X{]}^{\ell }\cap {\mathbb{Z}}^{\ell }}{\tau }_k\left(F(\mathbf{x})\right)$$

$${\tau }_k(n)=\#\{(x_1,x_2,...,x_k)\in {\mathbb{N}}^k:n=x_1{x}_2...x_k\},$$

$$\sum _{|F(\mathbf{x})|\le x}{\tau }_k(\left|F(\mathbf{x})\right|),$$

$$\sum _{|F(\mathbf{x})|\le X}\tau (\left|F(\mathbf{x})\right|)=c_0{X}^{\frac{2}{3}}\log X+c_1{X}^{\frac{2}{3}}+O_{\epsilon ,F}(X^{\frac{9}{14}+\epsilon }),$$

$$\sum _{|F(\mathbf{x})|\le X}\tau (\left|F(\mathbf{x})\right|)=c_2{X}^{\frac{1}{2}}\log X+O_F(X^{\frac{1}{2}}\log \log X),$$

$$\text{sideset}{}^{\ast }{\sum }_{n_1^2+n_2^6\le x}{\tau }_3(n_1^2+n_2^6)=c{x}^{\frac{2}{3}}(\log x{)}^2+O\left(x^{\frac{2}{3}}(\log x{)}^{\frac{7}{4}}(\log \log x{)}^{\frac{1}{2}}\right),$$

$$\sum _{F(\mathbf{x})\le X}{\tau }_k(F(\mathbf{x}))=\sum _{m\le x}{\tau }_k(m)r_F(m)$$

$$\sum _{1\le m_1,m_2\le X}\tau \left(m_1^2+m_2^2\right)=c_4{X}^2\log X+c_5{X}^2+O_{\epsilon }(X^{\frac{3}{2}+\epsilon }),$$

$$\sum _{1\le m_1,m_2,m_3\le X}{\tau }_3\left(m_1^2+m_2^2+m_3^2\right)=c_6{X}^3{\log }^{2}X+c_7{X}^3\log X+c_8{X}^3+O_{\epsilon }(X^{\frac{11}{4}+\epsilon }),$$

$$T_{k,F}(X):=\sum _{\mathbf{x}\in [1,X{]}^{\ell }\cap {\mathbb{Z}}^{\ell }}{\tau }_k(F(\mathbf{x})),$$

$$F(\mathbf{x})={\mathbf{x}}^t{Q}_m\mathbf{x}+{\mathbf{b}}^t\mathbf{x}+c,$$

$$\{\begin{array}{l}{\min }_{\mathbf{x}\in B_{\ell }(X)}F(\mathbf{x})>0\\ {\Delta }_F=\det \left(Q_m^t+Q_m\right)\ne 0.\end{array}$$

$$N_F(X)=X^2\sum _{1\le i,j\le \ell }{a}_{ij}+X\sum _{1\le r\le \ell }{b}_r+c.$$

$$T_{k,F}(X)=\sum _{r=0}^{k-1}{H}_{k,r}(F){\int }_{[1,X{]}^{\ell }}(\log F(\mathbf{t}){)}^r\mathrm{d}\mathbf{t}+O_{\epsilon ,k,F}\left(X^{\ell -\frac{\ell -2}{\ell +2}\min \left(1,\frac{4}{k+1}\right)+\epsilon }\right)$$

$$H_{k,r}(F)=\frac{1}{r!}\sum_{t=0}^{k-r-1}\frac{1}{t!}\left({\frac{{\rm d}^{t}L(s;k,F)}{{\rm d}s^{t}}}\bigg{|}_{s=1}\right){\rm Res}\left((s-1)^{r+t}\zeta(s)^{k};s=1\right),$$

$$\sum _{\begin{array}{c}m\le x\\ m\equiv h(\mathrm{mod}q)\end{array}}{\tau }_k(m)=M_k(x;h,q)+O_k(x^{1-\frac{2}{k+1}+\epsilon }),$$

$$M_k(x;h,q)=\mathrm{Res}\left(\zeta (s{)}^k\frac{x^s}{s}{f}_k(q,\delta ,s);s=1\right)$$

$$f_k(q,\delta ,s)=\frac{1}{\phi (q/\delta ){\delta }^s}\prod _{p|(q/\delta )}{\left(1-\frac{1}{p^s}\right)}^k\sum _{\begin{array}{c}{d}_1{d}_2...d_k=\delta \\ d_1,d_2,...,d_k>0\end{array}}\prod _{i=1}^{k-1}\prod _{\begin{array}{c}p|({\prod }_{r=i+1}^k{d}_r)\\ (p,q/\delta )=1\end{array}}\left(1-\frac{1}{p^s}\right).$$

$$A_k(x;h,q)=\sum _{\begin{array}{c}{d}_1{d}_2...d_k=\delta \\ d_1,d_2,...,d_{\ell }>0\end{array}}\sum _{\begin{array}{c}{t}_i|{\prod }_{r=i+1}^k{d}_r\\ i=1,2,...,k\\ (t_1{t}_2...t_k,q/\delta )=1\end{array}}\mu (\mathbf{t})A_k\left(\frac{x}{\delta t_1{t}_2...t_k};\overline{t_1{t}_2...t_{k}h/\delta },q/\delta \right)$$

$$A_k(x;h,q)=M_k(x;h,q)+{\Delta }_k(x;h,q),$$

$$M_k(x;h,q)=\sum _{\begin{array}{c}{d}_1{d}_2...d_k=\delta \\ d_1,d_2,...,d_k>0\end{array}}\sum _{\begin{array}{c}{t}_i|{\prod }_{r=i+1}^k{d}_r\\ i=1,2,...,k\\ (t_1{t}_2...t_k,q/\delta )=1\end{array}}\mu (\mathbf{t})\frac{x}{\delta t_1{t}_2...t_k}{P}_k\left(\log \left(\frac{x}{\delta t_1{t}_2...t_k}\right),\frac{q}{\delta }\right)$$

$${\Delta }_k(x;h,q)=\sum _{\begin{array}{c}{d}_1...d_k=\delta \\ d_1,...,d_k>0\end{array}}\sum _{\begin{array}{c}{t}_i|{\prod }_{r=i+1}^k{d}_r\\ i=1,2,...,k\\ (t_1...t_k,q/\delta )=1\end{array}}\mu (\mathbf{t})\left(D_k\left(0;\overline{\left(\frac{t_1...t_{k}h}{\delta }\right)},\frac{q}{\delta }\right)+O\left(\frac{{\tau }_k\left(q/\delta \right)x^{\frac{k-1}{k+1}}{\log }^{k}x}{(\delta t_1...t_k{)}^{\frac{k-1}{k+1}}\log x}\right)\right).$$

$$x{P}_k(\log x,q)=\frac{1}{\phi (q)}\mathrm{Res}\left({\left(\zeta (s)\sum _{d|q}{d}^{-s}\mu (d)\right)}^k\frac{x^s}{s};s=1\right).$$

$$M_k(x;h,q)$$

$$=\mathrm{Res}\left(\frac{x^s}{s}\frac{\zeta (s{)}^k}{\phi (q/\delta ){\delta }^s}\prod _{p|(q/\delta )}{\left(1-\frac{1}{p^s}\right)}^k\sum _{\begin{array}{c}{d}_1{d}_2...d_k=\delta \\ d_1,d_2,...,d_k>0\end{array}}\sum _{\begin{array}{c}{t}_i|{\prod }_{r=i+1}^k{d}_r\\ i=1,2,...,k\\ (t_1{t}_2...t_k,q/\delta )=1\end{array}}\frac{\mu (\mathbf{t})}{{\left(t_1...t_k\right)}^s};s=1\right)$$

$$:=\mathrm{Res}\left(\zeta (s{)}^k\frac{x^s}{s}{f}_k(q,\delta ,s);s=1\right),$$

$$f_k(q,\delta ,s)$$

$$=\frac{1}{\phi (q/\delta ){\delta }^s}\prod _{p|(q/\delta )}{\left(1-\frac{1}{p^s}\right)}^k\sum _{\begin{array}{c}{d}_1{d}_2...d_k=\delta \\ d_1,d_2,...,d_k>0\end{array}}\prod _{i=1}^{k-1}\prod _{\begin{array}{c}p|{\prod }_{r=i+1}^k{d}_r\\ (p,q/\delta )=1\end{array}}\left(1-\frac{1}{p^s}\right).$$

$${\Delta }_k(x;h,q)$$

$${\ll }_k\left(q^{\frac{k-1}{2}+\epsilon }+x^{\frac{k-1}{k+1}+\epsilon }\right)\sum _{d_1...d_k=\delta }\tau (\delta {)}^{k-1}{\ll }_k{x}^{1-\frac{2}{k+1}+\epsilon }.$$

$$F_{k,a}(q,s)=\sum _{h\in {\mathbb{Z}}_q}e\left(-\frac{ah}{q}\right)f_k(q,\delta ,s).$$

$$\frac{{\mathrm{d}}^s{F}_k(q,1)}{\mathrm{d}{s}^r}{\ll }_k{q}^{-1+\epsilon }$$

Full text

[TABLE]

with the help of the circle method. Here $\tau_{k}(n)=\#\{(x_{1},x_{2},...,x_{k})\in\mathbb{N}^{k}:n=x_{1}x_{2}...x_{k}\}$ with $k\in\mathbb{Z}_{\geq 2}$ is the multidimensional divisor function.

1. Introduction

The multidimensional divisor functions are generalisations of the divisor function $\tau(n)=\sum_{d|n}1$, defined by

[TABLE]

and counting the number of ways that $n$ can be written as a product of $k$ positive integer numbers. Understanding the average order of $\tau_{k}(n)$, as it ranges over the values taken by polynomials is an important topic in analytic number theory. The behavior of $\tau_{k}(n)$ is far less than perfectly understood even for $k=3$. For example, so far there are no asymptotic formulas for the sum $\sum_{m\leq x}\tau_{3}(m^{2}+1)$. We considers the sum

[TABLE]

where $F({\bf x})\in\mathbb{Z}[x_{1},x_{2}]$ is a binary form. For $k=2$ and $F({\bf x})$ is an irreducible cubic form, Greaves [1] showed that there exists constants $c_{0},c_{1}\in\mathbb{R}$ with $c_{0}>0$ depending only on $F$, such that

[TABLE]

holds for any $\varepsilon>0$. If $F$ is an irreducible quartic form, Daniel [2] showed that

[TABLE]

where $c_{2}$ is a constant depending only on $F$. For more related works, see e.g. la Bretèche, Browning [3] and Browning[4]. However, if $k\geq 3$, this kind of problems will become more complicated. There are few results in this direction. For $\tau_{3}(n)$, Friedlander and Iwaniec [5] showed that

[TABLE]

where $c$ is a constant and $*$ means that $(n_{1},n_{2})=1$. If $F({\bf x})$ is a positive definite quadratic form with $\ell\geq 2$ variables, then it is easy to obtain the sum

[TABLE]

by classical results of quadratic form, where $r_{F}(m)=\#\{{\bf x}\in\mathbb{Z}^{\ell}:m=F({\bf x})\}$. For example, Yu [6] obtained the following

[TABLE]

Sun and Zhang [7] obtained the following

[TABLE]

where $c_{4},c_{5},c_{6},c_{7},c_{8}$ are constants and $\varepsilon$ is an arbitrarily positive number. However this method does not working if $F$ is an indefinite quadratic form. On the other hand, nothing of the following sum

[TABLE]

is known for $k\geq 4$ and quadratic form $F$ is positive definite or indefinite.

The purpose of this paper is to investigate general problem as above. More precisely, let $F({\bf x})$ be a quadratic polynomial with $\ell(\geq 3)$ variables $x_{1},x_{2},..,x_{\ell}$ and integer coefficients. The vector ${\bf x}=(x_{1},x_{2},...,x_{\ell})^{t}\in\mathbb{Z}^{\ell}$ and denote $B_{\ell}(X)=[1,X]^{\ell}\cap\mathbb{Z}^{\ell}$ as a box for some sufficiently large positive number $X$. Also assume quadratic polynomial $F({\bf x})$ satisfies

[TABLE]

where $Q_{m}\in{\rm M}_{\ell}(\mathbb{Z})$ is an $\ell\times\ell$ matrix with entries $a_{ij}$, vector $\mathbf{b}=(b_{1},...,b_{\ell})^{t}\in\mathbb{Z}^{\ell}$, $c\in\mathbb{Z}$ and suppose those coefficients satisfy the following

[TABLE]

Thus $F({\bf x})$ has a maximum value $N_{F}(X)$ in the box $B_{\ell}(X)$ when $X$ is sufficiently large, say

[TABLE]

Our main result is the following.

Theorem 1.1**.**

Let $F$, $B(X)$ be defined as above, $k\geq 2$ and $\ell\geq 3$. For any $\varepsilon>0$ there exist constants $H_{k,0}(F)$, $H_{k,1}(F)$,…, and $H_{k,k-1}(F)$, such that

[TABLE]

and

[TABLE]

where the function $L(s;k,F)$ is given in Lemma 4.1.

Notation

The symbols $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ denote the positive integers, the integers and the real numbers, respectively. $e(z)=e^{2\pi iz}$, the letter $p$ always denotes a prime, $M^{t}$ is transpose operation of matrix $M$. The symbol $\mathbb{Z}_{q}$ represents shorthand for the groups $\mathbb{Z}/q\mathbb{Z}$. Also, the shorthand for the multiplicative group reduced residue classes $(\mathbb{Z}/q\mathbb{Z})^{*}$ is $\mathbb{Z}_{q}^{*}$. Occasionally we make use of the $\varepsilon$-convention: whenever $\varepsilon$ appears in a statement, it is asserted that the statement is true for all real $\varepsilon$. This allows us to write $x^{\varepsilon}\log x\ll x^{\varepsilon}$ , for example.

2. Primaries

The primary technique used in the proof of the main theorem is the circle method. We shall need the following results which need in the sequel. Lemma 2.4, Lemma 2.5 and Lemma 2.6 will be used in the estimates of the major arcs of the circle method. Lemma 2.8 will be used in the estimate of the minor arcs. To obtain Lemma 2.4, we firstly need the follows lemma.

Lemma 2.1** (R. A. Smith).**

Let $1\leq h\leq q$, $(q,h)=\delta$. Then for $q\leq x^{\frac{2}{k+1}}$ , we have

[TABLE]

where

[TABLE]

with

[TABLE]

Proof.

This lemma is essentially made by Smith [8], and we just change the form as needed. Firstly, by the equation (30) of [8], we get

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where the notations be followed. Theorem 3 of this paper yields

[TABLE]

where

[TABLE]

and

[TABLE]

By the definition of $P_{k}(\log x,q)$, namely (13),(21) and relatively talking about (21) of [8]. It is easily seen that

[TABLE]

Hence we obtain that

[TABLE]

where

[TABLE]

Smith [8] conjectured the validity of the estimate $D(0,h,q)\ll q^{\frac{k-1}{2}+\varepsilon}$ for any $(q,h)=1$ and proved by Matsumoto [9]. Which implies the bound

[TABLE]

This completes the proof of the lemma. ∎

We have the proposition which will be used in the proof of Lemma 2.4.

Proposition 2.2**.**

Let $q\geq 1$ be an integer, $(a,q)=1$ and denote $\delta=(h,q)$. Also let $f(q,\delta,s)$ be defined as in Lemma 2.1. Define

[TABLE]

Then $F_{k,a}(q,s)$ is independent of $a$ and we may write it as $F_{k}(q,s)$. Furthermore, $F_{k}(q,s)$ is multiplicative function and

[TABLE]

holds for any integer $r=0,1,...,k-1$.

Proof.

First, we have

[TABLE]

where $c_{\delta}(a)$ is the Ramanujan’s sum and the fact that if $(a,\delta)=1$ then $c_{\delta}(a)=\mu(\delta)$ be used. This result yields $F_{k,a}(q,s)$ independent on $a$. Suppose that positive integers $q_{1}$ and $q_{2}$ are coprime, then

[TABLE]

hence we just need to show

[TABLE]

whenever $\delta_{1}|q_{1}$ and $\delta_{2}|q_{2}$. It is obtained by the definition of $f_{k}(q,q/\delta,s)$, say

[TABLE]

Furthermore,

[TABLE]

where $\sigma={\rm Re}(s)$. It is easily seen that if $s=1+\rho e(\theta)$ with $\theta\in[0,1)$, then

[TABLE]

Thus we have

[TABLE]

On the other hand

[TABLE]

Therefore

[TABLE]

It is obviously that $F_{k}(q,s)$ is analytic in ${\mathbb{C}}$ for every $q$ which concerned. Hence one can use Cauchy estimate, say

[TABLE]

where $\rho\in(0,1)$. Hence combining with (2.2), we obtain that

[TABLE]

Thus complete the proof of the lemma. ∎

To apply the circle method, we need the following propositions.

Proposition 2.3**.**

Let $\alpha=a/q+\beta$ with $q\leq X^{\frac{2}{k+1}}$ be an positive integer and $(a,q)=1$. Define

[TABLE]

Then

[TABLE]

where $F_{k}(q,s)$ defined as Proposition 2.2.

Proof.

First, by Proposition 2.3 we have

[TABLE]

On the other hand,

[TABLE]

where $\delta=(q,h)$ and using Proposition 2.2 we complete the proof of the lemma. ∎

The Riemann zeta function is meromorphic with a single pole of order one at $s=1$. It can therefore be expanded as a Laurent series about $s=1$, say

[TABLE]

where

[TABLE]

are the Stieltjes constants. Therefore there exists constants $\alpha_{k,1}$, $\alpha_{k,2}$,…,$\alpha_{k,k}$ and a holomorphic function $h_{k}(s)$ on ${\mathbb{C}}$ such that

[TABLE]

Furthermore, we obtain that

[TABLE]

for any $x>0$, where $h_{k,x}(s)$ is a holomorphic function on ${\mathbb{C}}$ about $s$. On the other hand, we also have a Taylor series for $F_{k}(q,s)$ at $s=1$, say

[TABLE]

Therefore the residue of $\zeta(s)^{k}x^{s-1}F_{k}(q,s)$ at $s=1$ is

[TABLE]

We Define

[TABLE]

Then by Proposition 2.2 we have $\beta_{k,r}(q)\ll q^{-1+\varepsilon}$ and the results of Proposition 2.3 rewritten as

Lemma 2.4**.**

Let $\alpha=a/q+\beta$ with $q\leq X^{\frac{2}{k+1}}$ be an positive integer and $(a,q)=1$. Then, we have

[TABLE]

where

[TABLE]

with

[TABLE]

and where $f_{k}(q,\delta,s)$ defined by (2.1).

The following lemmas will be used in the estimate of the major arcs of the circle method.

Lemma 2.5**.**

Let $\alpha=a/q+\beta$ with $q$ be an positive integer and $(a,q)=1$. Define

[TABLE]

Then

[TABLE]

where

[TABLE]

Proof.

Firstly,

[TABLE]

We shall prove

[TABLE]

which immediately yields the proof. For any $a,b\in\mathbb{R}$ with $a\ll_{F}1$ and $b\ll_{F}X$, let us consider the follows estimate

[TABLE]

which obtained by partial integration directly. So above applied successively for each variables $x_{i}\equiv h_{i}~{}(\mathop{\rm{mod}}q)$ yields (2.7). ∎

Lemma 2.6**.**

Let $S_{F}(q,a)$ be defined as in Lemma 2.5 with $(q,a)=1$. Then for any $F({\bf x})$ be defined as above, we have

[TABLE]

Proof.

Firstly,

[TABLE]

It is easily seen that

[TABLE]

Hence we deduce that

[TABLE]

Since $({Q_{m}+Q_{m}^{t}})$ is nonsingular, hence

[TABLE]

This completes the proof. ∎

To give a good estimate for $I_{F}(\alpha,X)$ in the minor arcs of the circle method, we need the following lemmas.

Lemma 2.7**.**

Let $A\in M_{\ell}(\mathbb{Z})$ be a nonsingular matrix with column vectors ${\bf a}_{1},...,{\bf a}_{\ell}$. Also let let $\alpha=a/q+\beta$ with $q$ be an positive integer, $(a,q)=1$ and $|\beta|\leq q^{-2}$. Define

[TABLE]

where $B(X)=[1,X]^{\ell}\cap\mathbb{Z}^{\ell}$. Then

[TABLE]

Proof.

Firstly we have

[TABLE]

For the inner sum above

[TABLE]

notes that $\alpha=a/q+\beta$, then

[TABLE]

On the other hand, for each $v=1,2,...,\ell$, there exists some $t_{v}\in[0,1-1/q)$ and uniquely ${\bf h}^{\prime}=(h_{1}^{\prime},...,h_{\ell}^{\prime})^{t}\in\mathbb{Z}^{\ell}$ such that

[TABLE]

namely

[TABLE]

In this case, it has uniformly ${\bf a}_{v}^{t}{\bf h}q\beta\ll_{A}1$ for all ${\bf h}\in(\mathbb{Z}_{q})^{\ell}$, which implies the number of integers on above interval bounded by $O_{A}(1)$. For each $v$, let $N_{v}$ be an integer of above interval. If there exists an ${\bf h}\in(\mathbb{Z}_{q})^{\ell}$ such that ${\bf a}_{v}^{t}{\bf h}-qh_{v}^{\prime}=N_{v}$ for all $v=1,2,...,\ell$, namely

[TABLE]

then $(a,q)=1$ and $A$ nonsingular implies

[TABLE]

Hence the number of ${\bf h}\in(\mathbb{Z}_{q})^{\ell}$ satisfying (2.10) be bounded by $O_{A}(1)$. Furthermore, for all $(t_{1},t_{2},...,t_{\ell})\in[0,1-1/q)^{\ell}$, the number of ${\bf h}\in(\mathbb{Z}_{q})^{\ell}$ satisfy the condition

[TABLE]

bounded by $O_{A}(1)$. On the other hand $\parallel{\bf a}_{v}^{t}(q{\bf x}+{\bf h})\alpha\parallel\in\left[t_{v},t_{v}+1/q\right]$ if and only if

[TABLE]

Hence for all $(t_{1},t_{2},...,t_{\ell})\in[0,1-1/q)^{\ell}$, the number of ${\bf h}\in(\mathbb{Z}_{q})^{\ell}$ satisfying $\parallel{\bf a}_{v}^{t}(q{\bf x}+{\bf h})\alpha\parallel\in\left[t_{v},t_{v}+1/q\right]$ by bounded by $O_{A}(1)$.

For the convenience of discussion, $\forall s_{1},s_{2},...,s_{\ell}\in[0,q/2)\cap\mathbb{Z}$, we denote

[TABLE]

and

[TABLE]

Then

[TABLE]

Thus the sum (2.8) can be rewritten as

[TABLE]

Therefore we obtain that

[TABLE]

which completes the proof of the lemma. ∎

By this lemma, we have a nontrivial estimate for $I_{F}(\alpha,X)$ as follows.

Lemma 2.8**.**

Let $F({\bf x})$ defined by (1.2) and (1.3). Also let let $\alpha=a/q+\beta$ with $q$ be an positive integer, $(a,q)=1$ and $|\beta|\leq q^{-2}$. Then, we have

[TABLE]

Proof.

First of all, we have that

[TABLE]

Now we write the symmetric matrix ${Q_{mS}}=({\bf Q}_{1},{\bf Q}_{2},..,{\bf Q}_{\ell})^{t}~{}({\bf Q}_{j}\in\mathbb{Z}^{\ell},j=1,...,\ell)$. Then, using the fact that

[TABLE]

we obtain that

[TABLE]

Using the same method as in the proof of Lemma 2.8, we can derive that

[TABLE]

This completes the proof of the lemma. ∎

3. Singular integral

The well known results says that the gaussian integral

[TABLE]

converges if $A$ is a symmetric complex matrix with the real part of $A$ is non-negative and no eigenvalue $\alpha_{i}$ of $A$ vanishes. Hence we obtain that

[TABLE]

where ${\rm sign}(\lambda)$ is general symbol function and $\lambda\neq 0$. We have the following lemma.

Lemma 3.1**.**

We have

[TABLE]

Proof.

Firstly, we have

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where $F({\bf t},X)={\bf t}^{t}Q_{m}{\bf t}+{{\bf b}^{t}}{\bf t}/{X}+{c}/{X^{2}}$. If $|\mu|>Q/q\geq 1$, using (3) then

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The above result implies that

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On the other hand,

[TABLE]

where we have used the fact: $\int_{0}^{\infty}\frac{\sin(\alpha x)}{x}\mathrm{d}x=\frac{\pi}{2}{\rm sign}(\alpha)$. Note that

[TABLE]

then part integration yields

[TABLE]

Together with (3.2) and above, we get the proof the lemma. ∎

4. The proof of main theorem

Here we refer the methods of Pleasants [10] to deal with the minor arcs. Firstly, let $j\in\mathbb{N}$ and define

[TABLE]

It is obviously that,

[TABLE]

and $\mathfrak{M}(2^{j}Q)=[0,1]$ when $j>\lfloor(\log(X/Q))/(\log 2)\rfloor:=N$ by well know Dirichlet’s approximation theorem. If we define

[TABLE]

for $j=0,1,2,...,N$, then for all $i\neq j~{}(i,j=0,1,2,..,N)$ one has

[TABLE]

We take $\mathfrak{M}(Q)$ as the major arcs, and the minor arcs is $\mathfrak{m}(Q)=[0,1]\setminus\mathfrak{M}(Q)$. As we all know, $\mathfrak{M}(2^{j}Q)$ is the union of all disjoint small intervals $\mathfrak{M}_{j}(q,a)$ with $1\leq q\leq 2^{j}Q$ and $(a,q)=1$, where $\mathfrak{M}_{j}(q,a)=[{a}/{q}-2^{j}Q({qX^{2}})^{-1},{a}/{q}+2^{j}Q({qX^{2}})^{-1}]$. Thus we have

[TABLE]

for all $j=0,1,...,N$ and $\mathfrak{m}(Q)=\bigcup_{j=0}^{N}\mathcal{F}_{j}(Q)$. Therefore

[TABLE]

Applying the Cauchy-Schwarz inequality give an estimate for the minor arcs integral as follows

[TABLE]

where $\left|\mathcal{F}_{j}(Q)\right|$ is the Lebesgue measure of the set $\mathcal{F}_{j}(Q)$. By (4.1) one has

[TABLE]

For $j\leq N$, notes that $2^{j}Q\leq X$ and Lemma 2.8 one has

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Hence by (4.3), (4.4) and $\ell\geq 3$ implies

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On the other hand

[TABLE]

hence together it with (4) and (4) we obtain that

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For the major arc, by Lemma 2.4 and Lemma 2.5, we have

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Note that Lemma 2.6 and $\beta_{k,r}\ll_{k}q^{-1+\varepsilon}$, we obtain that

[TABLE]

where

[TABLE]

On the other hand, by Lemma 3.1 we have

[TABLE]

where

[TABLE]

It is easily seen that when $Q=X^{\min(1,{4}/{(k+1)})/{(\ell+2)}}$, one has the optimal estimate

[TABLE]

We define

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then combine Lemma 2.4 and (4.8) we obtain that

[TABLE]

We next try to give an explicit expression for $L(s;k,F)$.

Lemma 4.1**.**

The function $L(s;k,F)$ has the Euler product as follows

[TABLE]

where

[TABLE]

[TABLE]

for $n\in\mathbb{N}_{\geq 1}$ and

[TABLE]

Proof.

It is easily seen that

[TABLE]

is real and multiplicative. When $q=p^{m}$ is a prime power with integer $m\geq 1$. It is easily seen that

[TABLE]

On the other hand, by Lemma 2.2 we shown that $F_{k}(q,s)$ is also multiplicative. Thus above implies the Euler product of $L(s;k,F)$. Applying Lemma 2.1 and Proposition 2.2, we show that

[TABLE]

For the first term above, denote by

[TABLE]

Then clearly for $m\geq 1$,

[TABLE]

[TABLE]

Hence

[TABLE]

∎

Combining above estimates and calculations, we obtain the proof of the main theorem.