An asymptotic formula for the number of prime solutions for multivariate linear equations

Abstract

In this paper, we study the multivariate linear equations with arbitrary positive integral coefficients. Under the Generalized Riemann Hypothesis, we obtained the asymptotic formula for the linear equations with more than five prime variables. This asymptotic formula is composed of three parts, that is, the first main term, the explicit second main term and the error term. Among them, the first main term is similar with the former one, the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions, and our error term improves the former one.

Keywords

Prime variables / linear equations / circle methods / estimation of error terms

1 Introduction

In this paper we are concerned with the following linear equations

(1)\begin{array}{r}{a}_1{p}_1+a_2{p}_2+\cdots +a_k{p}_k=b,(k⩾3),\end{array}

where p_1, \cdots ,p_k are prime varibales and the coefficients a_1, \cdots ,a_k are arbitrary positive integers for which

(2)\begin{array}{r}(a_1,a_2,\cdots ,a_k)=1,\end{array}

b is given sufficently large positive integer satisfying two conditions:

(3)

(4)\begin{array}{r}{a}_1+a_2+\cdots +a_k\equiv b\mathrm{mod}2.\end{array}

It is known that when k=2 and a_1=a_2=1, equation (1) is the famous even Goldbach problem which is an open problem; when k=3, a_1=a_2 =a_3= 1 and sufficently large b, it is the odd Goldbach problem which is solved by Vinogradov in 1937 [8]. Moreover, he showed an asymptotic formula for the number of prime solutions:

\begin{array}{r}{r}_k(n)=\frac{n^{k-1}}{(k-1)!}{\mathfrak{G}}_k(n)+O(n^{k-1}(\log n{)}^{-A}),(A>0),\end{array}

where

\begin{array}{r}{r}_k(n)=\sum _{p_1+p_2+\cdots +p_k=n}(\log p_1)(\log p_2)\cdots (\log p_k),\end{array}

and the singular series

\begin{array}{r}{\mathfrak{G}}_k(n)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (q)}{\varphi (q{)}^k}\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(\frac{-an}{q}\right),\end{array}

where e(u)={\mathrm{e}}^{2\pi \mathrm{i}u}.

In 1973, Montgomery and Vaughan [5] showed that if we define

\begin{array}{r}\theta =sup\{\beta :L(\beta +\mathrm{i}\gamma ,\chi )=0\},\end{array}

where the supremum is over all zeros of all the L-functions for Dirichlet characters, then the error term of the asymptotic formula must be

(5)\begin{array}{r}O(n^{k-2+\theta +ϵ})(ϵ>0).\end{array}

Then in 1997, Friedlander and Goldston [1] using the upper bound estimate of zeros of L(s,\chi ) in \mathrm{\Re }s⩾\sigma, , that is N(\sigma ,T,\chi ), found the estimate of error term similarly with (5) as following

\begin{array}{r}O(n^{k-2+\theta +ϵ})+O(n^{{\eta }_k+ϵ}),\end{array}

where {\eta }_3=\frac{9}{5}, {\eta }_4=\frac{13}{5}, and {\eta }_k =0(k⩾5).

In 2012, assuming the Generalized Riemann Hypothesis (GRH), Languasco and Zaccagnini [2] considered

\begin{array}{r}{R}_k(n)=\sum _{n_1+n_2+\cdots +n_k=n}\mathrm{\Lambda }(n_1)\mathrm{\Lambda }(n_2)\cdots \mathrm{\Lambda }(n_k),(k⩾5),\end{array}

and gave an explicit asymptotic formula

\begin{array}{rl}{R}_k(n)=& \frac{n^{k-1}}{(k-1)!}-k\sum _{q=1}^{\mathrm{\infty }}\frac{\mu (q{)}^{k-1}}{\varphi (q{)}^k}\sum _{\chi \mathrm{mod}q}{C}_{\chi }(-n)\tau (\overline{\chi })\\ & \times \sum _{\rho }\frac{n^{\rho +k-2}}{\rho (\rho +1)\cdots (\rho +k-2)}+O(n^{k-\frac{7}{4}}(\log n{)}^{k-1}),\end{array}

where in the error term, \frac{7}{4} can be insteaded by 2 when k ⩾6, and (\log n{)}^{k-1} can be instead by (\log n{)}^2 when k ⩾7.

Inspired by the above results, in this paper, we study the number of prime solutions for more general linear equations (1), that is

\begin{array}{r}{I}_k(b)=\sum _{a_1{n}_1+a_2{n}_2+\cdots +a_k{n}_k=b}\mathrm{\Lambda }(n_1)\mathrm{\Lambda }(n_2)\cdots \mathrm{\Lambda }(n_k),(k⩾5),\end{array}

and show the asymptotic formula for I_k(b) assuming the GRH as the following theorem.

Theorem 1　 Assuming the GRH, for any sufficently large positive integer b which satisfies condition (2)−(4), we have

\begin{array}{rl}{I}_k(b)=& \frac{n^{k-1}}{(k-1)!A_k}{G}_k(b)+\sum _{j=1}^k{T}_j(b)\sum _{\rho }\frac{b^{\rho +k-2}{a}_j^{1-\rho }}{\rho (\rho +1)\cdots (\rho +k-2)A_k}\\ & +O(B^2{b}^{k-\frac{7}{4}}(\log b{)}^{k-1}),\end{array}

where k⩾5, A_k=a_1 a_2\cdots a_k, B=\underset{1⩽i⩽k}{max}\{a_i\},

\begin{array}{r}{G}_k(b)=\sum _{q=1}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^k}{C}_q(-b)\prod _{j=1}^k{C}_q(a_j),\end{array}

and

\begin{array}{r}{T}_j(b)=\sum _{q=1}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^k}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{C}_q(a_i)\sum _{\chi \mathrm{mod}q}{C}_{\chi }(a_j)C_{\chi }(-b).\end{array}

We note that in 2004, Li Weiping and Zhou Haigang [3] also studied the asymptotic formula for the number of solutions to the linear equation with integral coefficients of prime variables, and the error term of their asymptotic formula is O(N^{k-\frac{3}{2}}) (k ⩾5) provided that the generalized Riemann hypothesis holds. In contrast, the advantage of the asymptotic formula in Theorem 1 is that adding a second principal term not only makes the remainder smaller, but also makes the asymptotic formula more precise. It should also be noted that Vaughan and Wooley [7] gave a higher-order asymptotic formula for the number of solutions to Waring's problem in 2016, so does the Waring-Goldbach problem also have a corresponding higher-order asymptotic formula for the number of prime solutions? The results of this article can also be seen as a partial answer to this question.

2 Preparation and outline of the method

As usual, we denote the Euler function, the von-Mangoldt function, the Ramanujan function and the Gauss function by \varphi (n), \mathrm{\Lambda }(n), C_q(n), and C_{\chi }( n), respectively. Let ϵ be sufficiently small positive numbers which may not be equal everywhere in the paper.

According to the Dirichlet approximation theorem, we define the Farey fractions of level Q with 1 ⩽Q ⩽b,

\left\{\frac{a}{q}:1⩽a⩽q,(a,q)=1\right\}.

The Farey arcs centered at a/q is

{\mathcal{M}}_{q,a}=\left(\frac{a+a^{\prime }}{q+q^{\prime }},\frac{a+a{}^{″}}{q+q{}^{″}}\right],\left(\frac{a}{q}\ne \frac{1}{1}\right),

and {\mathcal{M}}_{1,1}=(1-\frac{1}{Q+1} ,1+\frac{1}{Q+1} ]. Here are three consecutive Farey fractions. Let

{\xi }_{q,a}=\left(-\frac{1}{q(q+q^{\prime })},\frac{1}{q(q+q{}^{″})}\right], \left(\frac{a}{q}\ne \frac{1}{1}\right),

and {\xi }_{1,1}=(-\frac{1}{Q+1} ,\frac{1}{Q+1}]. Then any \alpha \in (\frac{1}{Q+1},1+\frac{1}{Q+1}] can be represented by

\begin{array}{r}\alpha =\frac{a}{q}+\eta ,\text{where}1⩽a⩽q⩽Q,(a,q)=1\text{and}\eta \in {\xi }_{q,a}.\end{array}

Moreover, we have (- \frac{1}{2qQ},\frac{1}{2qQ})\subseteq {\xi }_{q,a}\subseteq (-\frac{1}{qQ},\frac{1}{qQ}).

Set the weighted exponential sum as

\begin{array}{r}{S}_j(\alpha )=\sum _{n=1}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha ).\end{array}

In the following, applying the orthogonality of the Dirichlet character, we divide S_j( \alpha ) into three parts.

Firstly, according to the definition of von-Mangoldt function, we know that for any 1⩽q⩽Q,

\begin{array}{r}{S}_j(\alpha )=\sum _{\genfrac{}{}{0ex}{}{n=1}{(n,q)=1}}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha )+\sum _{\genfrac{}{}{0ex}{}{n=1}{q\mid n}}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha ).\end{array}

Here

\begin{array}{r}\sum _{\genfrac{}{}{0ex}{}{n=1}{q\mid n}}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha )=\sum _{p\mid q}\sum _{t=1}^{\mathrm{\infty }}\mathrm{\Lambda }(p^t){\mathrm{e}}^{-\frac{a_j{p}^t}{b}}e(a_j{p}^t\alpha )\ll \sum _{p\mid q}\log p\sum _{t=1}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{a_j{p}^t}{b}}.\end{array}

Since

\begin{array}{r}\sum _{t=1}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{a_j{p}^t}{b}}=\sum _{t⩽\log b}1+\sum _{t>\log b}\frac{b}{a_j{p}^t}\ll \log b+\sum _{t⩾\log b}\frac{b}{a_j{\mathrm{e}}^{\log b}(\frac{p}{e}{)}^t}\ll \log b,\end{array}

we have

\begin{array}{r}{S}_j(\alpha )=\sum _{\genfrac{}{}{0ex}{}{n=1}{(n,q)=1}}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha )+O(\log b\log q).\end{array}

Secondly,

\begin{array}{r}\sum _{\genfrac{}{}{0ex}{}{n=1}{(n,q)=1}}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha )=\sum _{\genfrac{}{}{0ex}{}{t=1}{(t,q)=1}}^{q}e(a_j\frac{t}{q})\sum _{\genfrac{}{}{0ex}{}{n=1}{n\equiv a^{-1}t\mathrm{mod}q}}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\eta ),\end{array}

where a{a}^{-1}\equiv 1\mathrm{mod}q. Applying the orthogonality of the Dirichlet character, the above equation is

\begin{array}{r}\sum _{\genfrac{}{}{0ex}{}{n=1}{(n,q)=1}}^{\mathrm{\infty }}\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\alpha )=\frac{1}{\varphi (q)}\sum _{\chi \mathrm{mod}q}\chi (a)C_{\chi }(a_j){\tilde{W}}_j(\chi ,\eta ),\end{array}

where

\begin{array}{r}{\tilde{W}}_j(\chi ,\eta )=\sum _{n=1}^{\mathrm{\infty }}\overline{\chi }(n)\mathrm{\Lambda }(n){\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\eta ).\end{array}

Thirdly, let

\begin{array}{r}{V}_j(\eta )=\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{a_{j}n}{b}}e(a_{j}n\eta ),\end{array}

\begin{array}{r}{W}_j(\chi ,\eta )=\{\begin{array}{ll}{\tilde{W}}_j(\chi ,\eta )-V_j(\eta ),& \chi ={\chi }_0,\\ {\tilde{W}}_j(\chi ,\eta ),& \chi \ne {\chi }_0,\end{array}\end{array}

where {\chi }_0 is the principal character. Then

\begin{array}{rl}{S}_j(\alpha )& =\frac{C_q(a_j)}{\varphi (q)}{V}_j(\eta )+\frac{1}{\varphi (q)}\sum _{\chi \mathrm{mod}q}\chi (a)C_{\chi }(a_j)W_j(\chi ,\eta )+O(\log b\log q)\\ & =:S_{j,1}+S_{j,2}+S_{j,3}.\end{array}

In this way, we have

\begin{array}{rl}\prod _{j=1}^k{S}_j(\alpha )& =\prod _{i=1}^k{S}_{j,1}+\sum _{j=1}^k\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{S}_{i,1}(S_{j,2}+S_{j,3})+O\left(\sum _{m=0}^{k-2}\prod _{i=1}^m{S}_{i,1}\prod _{j=m+1}^k(S_{j,2}+S_{j,3})\right)\\ & =:S_1+S_2+S_3,\end{array}

and

(6)\begin{array}{rl}{I}_k(b)=& {\int }_{\frac{1}{1+Q}}^{1+\frac{1}{Q+1}}\prod _{j=1}^k{S}_j(\alpha )e(-b\alpha )\mathrm{d}\alpha \\ =& \sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{{\xi }_{q,a}}(S_1+S_2+S_3)e(-b\eta )\mathrm{d}\eta \\ =:& M_{k,1}(b)+M_{k,2}(b)+M_{k,3}(b),\end{array}

where

\begin{array}{r}{M}_{k,i}(b)=\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{{\xi }_{q,a}}{S}_{i}e(-b\eta )\mathrm{d}\eta ,(i=1,2,3).\end{array}

The latter sections of this article are arranged as follows. In Section 2, we estimate M_{k,1}(b), and give the first main term of the asymptotic formula for I_k( b). In Section 3, we estimate M_{k,2}(b), and give the explicit second main term of the asymptotic formula. In Section 4, we estimate M_{k,3}(b), and give the error term of the asymptotic formula and the proof of Theorem 1.

3 Estimate {\mathit{M}}_{\mathit{k}\mathbf{,}\mathbf{1}}\mathbf{(}\mathit{b}\mathbf{)}

Let

\begin{array}{r}{z}_j=\frac{a_j}{b}-2\pi \mathrm{i}{a}_j\eta =a_j\left(\frac{1}{b}-2\pi \mathrm{i}\eta \right).\end{array}

Then

\begin{array}{r}{V}_j(\eta )=\sum _{n=1}^{\mathrm{\infty }}{e}^{-\frac{a_{j}n}{b}}e(a_{j}n\eta )=z_j^{-1}+O(1).\end{array}

Therefore

(7)\begin{array}{r}{\int }_{{\xi }_{q,a}}\prod _{j=1}^k{V}_j(\eta )e(-b\eta )\mathrm{d}\eta ={\int }_{{\xi }_{q,a}}\frac{e(-b\eta )}{z_1{z}_2\cdots z_k}\mathrm{d}\eta +O\left({\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}\frac{\mathrm{d}\eta }{|z_1{z}_2\cdots z_{k-1}|}\right),\end{array}

where

\begin{array}{rl}{\int }_{q,a}\frac{e(-b\eta )}{z_1{z}_2\cdots z_k}\mathrm{d}\eta =& {\int }_{{\xi }_{q,a}}{\displaystyle \frac{e(-b\eta )}{a_1{a}_2\cdots a_k{\left({\displaystyle \frac{1}{b}}-2\pi \mathrm{i}\eta \right)}^k}}\mathrm{d}\eta \\ =& \frac{b^{k-1}}{A_k(k-1)!}+O((qQ{)}^{k-1}).\end{array}

Here A_k=a_1{a}_2 \cdots a_k, which can be deduced by Lemma 8 of [2]. In the same way, we know the error term of equation (7) is O(b^{k-2}). Hence, we have

(8)\begin{array}{r}{\int }_{{\xi }_{q,a}}\prod _{j=1}^k{v}_j(\eta )e(-b\eta )\mathrm{d}\eta =\frac{b^{k-1}}{A_k(k-1)!}+O((qQ{)}^{k-1}+b^{k-2}).\end{array}

If we put equation (8) into M_{k,1}(b), and let

(9)\begin{array}{r}{G}_k(b)=\sum _{q=1}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^k}{C}_q(-b)\prod _{j=1}^k{C}_q(a_j),\end{array}

then we have

(10)\begin{array}{rl}{M}_{k,1}(b)=& \frac{b^{k-1}}{A_k(k-1)!}{G}_k(b)+O\left(b^{k-1}\sum _{q>Q}\frac{1}{\varphi (q{)}^k}{C}_q(-b)\prod _{j=1}^k{C}_q(a_j)\right)\\ & +O\left(Q^{k-1}\sum _{q=1}^Q\frac{q^{k-1}}{\varphi (q{)}^k}{C}_q(-b)\prod _{j=1}^k{C}_q(a_j)\right)\\ & +O\left(b^{k-2}\sum _{q=1}^Q\frac{1}{\varphi (q{)}^k}{C}_q(-b)\prod _{j=1}^k{C}_q(a_j)\right).\end{array}

Here G_k( b) is the singular series. Under conditons (2)−(4), we can show that G_k(b) is convergent and positive [4]. In fact, since [6]

\begin{array}{r}{C}_q(n)=\mu \left(\frac{q}{(n,q)}\right)\varphi (q){\varphi }^{-1}\left(\frac{q}{(n,q)}\right),\end{array}

under conditons (2)−(4), we have

\begin{array}{rl}{G}_k(b)=& \sum _{n=1}^{\mathrm{\infty }}\frac{\mu \left({\displaystyle \frac{q}{(b,q)}}\right)\varphi (q){\displaystyle \prod _{j=1}^k\mu \left({\displaystyle \frac{q}{(a_j,q)}}\right)\varphi (q)}}{\varphi (q{)}^k\varphi \left({\displaystyle \frac{q}{(b,q)}}\right){\displaystyle \prod _{j=1}^k\varphi \left({\displaystyle \frac{q}{(a_j,q)}}\right)}}\\ \ll & \sum _{q=1}^{\mathrm{\infty }}\frac{\mu (q{)}^2}{\varphi (q){\displaystyle \prod _{j=1}^{k-1}\varphi (\frac{q}{(a_j,q)})}}\\ \ll & \sum _{q⩽a_1{a}_2\cdots a_{k-1}}\frac{\mu (q{)}^2}{\varphi (q)}+\sum _{q>a_1{a}_2\cdots a_{k-1}}\frac{\mu (q{)}^2{a}_1{a}_2\cdots a_{k-1}}{\varphi (q{)}^k}\\ \ll & \sum _{q⩽a_1{a}_2\cdots a_{k-1}}\frac{\mu (q{)}^2}{\varphi (q)}+\sum _{q>a_1{a}_2\cdots a_{k-1}}\frac{\mu (q{)}^2}{\varphi (q{)}^{k-1}}=O_{B,k}(1),\end{array}

where k⩾3, and

\begin{array}{r}B=\underset{1⩽i⩽k}{max}\{a_i\}.\end{array}

Therefore G_k(b) is convergent. Moreover,

G_k(b) =1+\sum _{q⩾2}\frac{\mu \left({\displaystyle \frac{q}{(b,q)}}\right)\varphi (q){\displaystyle \prod _{j=1}^k\mu \left({\displaystyle \frac{q}{(a_j,q)}}\right)\varphi (q)}}{\varphi (q{)}^k\varphi \left({\displaystyle \frac{q}{(b,q)}}\right){\displaystyle \prod _{j=1}^k\varphi \left({\displaystyle \frac{q}{(a_j,q)}}\right)}}=:1+{\sum }^{\prime },

under equations (2)−(4),

Therefore, we have G_k(b) >0.

In the following we process three parts of error terms. The first error term of equation (10) is

\begin{array}{r}\ll b^{k-1}\sum _{q>Q}\frac{\mu (q{)}^2(b,q){\displaystyle \prod _{j=1}^k(a_j,q)}}{\varphi (q{)}^k}\ll b^{k-1}\sum _{q>Q}\sum _{d|q}\frac{d^k}{\varphi (q{)}^k}\ll b^{k-1}\sum _{q>Q}\frac{1}{\varphi (q{)}^{k-ϵ}}.\end{array}

Here we make use of \sum _{d|q}{d}^{k-1}\ll q^{ϵ}, when k⩾3, this error term is

\begin{array}{r}\ll b^{k-1}{Q}^{1-k}.\end{array}

Further, when Q=b^{1/2}/2, this error term is

(11)\begin{array}{r}\ll b^{k-1}{b}^{\frac{1}{2}-\frac{k}{2}}\ll b^{k-2}.\end{array}

In the same way, we have the second error term of equation (10) is

\begin{array}{r}\ll Q^{k-1}\sum _{q=1}^Q\frac{q^{k-1}}{\varphi (q{)}^k}\sum _{d|q}{d}^{k-1}\ll Q^{k-1}\sum _{q=1}^Q\frac{1}{\varphi (q{)}^{1-ϵ}}\ll Q^k.\end{array}

Further, when k⩾4 and Q=b^{1/2}/2, this second error term is

(12)\begin{array}{r}\ll b^{\frac{k}{2}}\ll b^{k-2}.\end{array}

In addtion, we have the third error term of equation (10) is

(13)\begin{array}{r}\ll b^{k-2}\sum _{q=1}^Q\frac{\sum _{d|q}{d}^{k-1}}{\varphi (q{)}^k}\ll b^{k-2}.\end{array}

Putting inequations (11)−(13) into the formula (10), we have the following lemma.

Lemma 1　 If k⩾4 and Q=b^{1/2}/2, and define G_k(b) by (9), then we have

\begin{array}{r}{M}_{k,1}(b)=\frac{b^{k-1}}{A_k(k-1)!}{G}_k(b)+O(b^{k-2}),\end{array}

where A_k=a_1 a_2\cdots a_k.

4 Estimate {\mathit{M}}_{\mathit{k}\mathbf{,}\mathbf{2}}\mathbf{(}\mathit{b}\mathbf{)}

We divide M_{k,2}(b) into two parts as

\begin{array}{rl}{M}_{k,2}(b)=& \sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{{\xi }_{q,a}}\sum _{j=1}^k\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{S}_{i,1}{S}_{j,2}e(-b\eta )\mathrm{d}\eta \\ & +\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{a}\right){\int }_{{\xi }_{q,a}}\sum _{j=1}^k\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{S}_{i,1}{S}_{j,3}e(-b\eta )\mathrm{d}\eta .\end{array}

Notice S_{j,3}=O (\log b\log q). Then we estimate the first part of M_{k,2}( b), and divide it into three parts again as

\begin{array}{r}{M}_{2,1}=\sum _{q=1}^{\mathrm{\infty }}\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{-\frac{1}{2}}^{\frac{1}{2}}\sum _{j=1}^k\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{S}_{i,1}{S}_{j,2}e(-b\eta )\mathrm{d}\eta ,\end{array}

\begin{array}{r}{M}_{2,2}=O\left(\sum _{q>Q}\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{-\frac{1}{2}}^{\frac{1}{2}}\sum _{j=1}^k\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{S}_{i,1}{S}_{j,2}e(-b\eta )\mathrm{d}\eta \right),\end{array}

\begin{array}{r}{M}_{2.3}=O\left(\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{\frac{1}{qQ}}^{\frac{1}{2}}\sum _{j=1}^k\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{S}_{i,1}{S}_{j,2}e(-b\eta )\mathrm{d}\eta \right).\end{array}

In the following, we shall estimate M_{2,1}, M_{2,2} and M_{2,3} one by one. According to Lemma 2 of [2], let x>2 be a real number, j ⩾1, q⩾1 be integers, \chi is the character modulo q, and define

Then we have:

Lemma 2　 Assume that the GRH holds for every L(s,\chi ). Then

\begin{array}{r}{\psi }_j(x,\chi )=\delta (\chi )\frac{x^{j+1}}{(j+1)!}-\sum _{\rho }\frac{x^{\rho +j}}{\rho (\rho +1)\cdots (\rho +j)}+O(x^j\log b\log q),\end{array}

where \rho is over all the non-trivial zeros of L(s,\chi ), and if \chi ={\chi }_0, then \delta ( \chi )=1, and \delta (\chi )=0, otherwise.

Applying Lemma 2, we have

(14)\begin{array}{rl}& {\int }_{-\frac{1}{2}}^{\frac{1}{2}}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{V}_i(\eta )W_j(\chi ,\eta )\mathrm{d}\eta \\ =& \sum _{m_j⩽\frac{b}{a_j}}(\mathrm{\Lambda }(m_j)\overline{\chi }(m_j)-\delta (\chi ))\left(\frac{a_j(b-a_j{m}_j{)}^{k-2}}{(k-2)!A_k}+O((b-a_j{m}_j{)}^{k-3})\right)\\ =& \frac{a_j^{k-1}}{A_k}{\psi }_{k-2}\left(\frac{b}{a_j},\chi \right)-\frac{b^{k-1}}{(k-1)!A_k}\delta (\chi )+O(b^{k-2})\\ =& \sum _{\rho }\frac{b^{k-2+\rho }{a}_j^{1-\rho }}{\rho (\rho +1)\cdots (\rho +k-2)A_k}+O(b^{k-2}\log b\log q).\end{array}

Set

(15)\begin{array}{rl}{T}_j(b)=& \sum _{q=1}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^k}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{C}_q(a_i)\sum _{\genfrac{}{}{0ex}{}{a=1}{(a.q)=1}}^{q}e\left(-b\frac{a}{q}\right)\sum _{\chi \mathrm{mod}q}\chi (a)C_{\chi }(a_j)\\ =& \sum _{q=1}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^k}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{C}_q(a_i)\sum _{\chi \mathrm{mod}q}{C}_{\chi }(a_j)C_{\chi }(-b).\end{array}

Then T_j( b) is convergent. It is because that

\begin{array}{r}{T}_j(b)⩽\sum _{\genfrac{}{}{0ex}{}{q=1}{q^{\ast }|q}}^{\mathrm{\infty }}\frac{\mu (q{)}^2}{\varphi (q{)}^k}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k(a_j,q)\varphi (q)q^{\ast }⩽\sum _{q=1}^{\mathrm{\infty }}\sum _{d|q}\frac{d^k}{\varphi (q{)}^{k-1}}\ll \sum _{q=1}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^{k-1-ϵ}}.\end{array}

Now M_{2,1} can be rewritten as

\begin{array}{r}{M}_{2,1}=\sum _{j=1}^k{T}_j(b)\sum _{\rho }\frac{b^{k-2+\rho }{a}_j^{1-\rho }}{\rho (\rho +1)\cdots (\rho +k-2)A_k}+O(b^{k-2}\log b).\end{array}

When estimate M_{2,2}, we apply (14) and

\begin{array}{r}\sum _{q>Q}\frac{1}{\varphi (q{)}^k}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{C}_q(a_i)\sum _{\chi \mathrm{mod}q}{C}_{\chi }(a_j)C_{\chi }(b)\ll \sum _{q>Q}\frac{1}{\varphi (q{)}^{k-1ϵ}}\ll Q^{1+ϵ-k}\end{array}

to deduce that when k⩾3 and Q=b^{1/2}/2,

\begin{array}{r}{M}_{2,2}\ll b^{k-1}{Q}^{1+ϵ-k}\ll b^{k-2}.\end{array}

At the same time, applying V_j(\eta )\ll qQ and W_j(\chi ,\eta )\ll b with \eta \in (\frac{1}{2qQ}, \frac{1}{2}), we know that when k⩾5 and Q=b^{1/2}/2,

\begin{array}{r}{M}_{2,3}\ll b{Q}^{k-1+ϵ}\ll b^{k-2+ϵ}.\end{array}

Then we estimtate the second part of M_{k,2}(b). Note that

\begin{array}{r}{\int }_{{\xi }_{q,a}}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{V}_i(\eta )\mathrm{d}\eta \ll {\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}\frac{1}{{\eta }^{k-1}}\mathrm{d}\eta \ll (qQ{)}^{k-2}.\end{array}

Therefore, when k⩾5 and Q=b^{1/2}/2, this part is

\begin{array}{r}\ll \sum _{q=1}^Q\frac{q^{k-2}}{\varphi (q{)}^k}\prod _{\genfrac{}{}{0ex}{}{i=1}{i\ne j}}^k{C}_q(a_i)\sum _{\chi \mathrm{mod}q}{C}_{\chi }(a_j)C_{\chi }(-b)\log q{Q}^{k-2}b\log b\ll b^{k-2}\log b.\end{array}

Hence we finish the estimate of M_{k,2}(b) and obtain the following lemma.

Lemma 3　 Assume the GRH, and when k⩾5 and Q=b^{1/2}/2, define T_j( b) as (15). Then we have

\begin{array}{r}{M}_{k,2}(b)=\sum _{j=1}^k{T}_j(b)\sum _{\rho }\frac{b^{k-2+\rho }{a}_j^{1-\rho }}{\rho (\rho +1)\cdots (\rho +k-2)A_k}+O(b^{k-2}\log b),\end{array}

where \rho is over all the non-trivial zeros of L(s,\chi ).

5 Estimate {\mathit{M}}_{\mathit{k}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{b}\mathbf{)}

First of all, when \chi \ne {\chi }_0,

\begin{array}{r}{W}_j(\chi ,\eta )=\sum _{n=1}^{\mathrm{\infty }}\overline{\chi }(n)\mathrm{\Lambda }(n){\mathrm{e}}^{-z_{j}n},\end{array}

where z_j=\frac{a_j}{b}-2\pi \mathrm{i}{a}_j\eta. Set \sigma >1, and let (\sigma ) denote the line with real part equal to \sigma in the complex plane. According to (1.2.5) and (6.3.25) in [6], we have

\begin{array}{r}{\mathrm{e}}^{-z_{j}n}=\frac{1}{2\pi \mathrm{i}}{\int }_{(\sigma )}(z_{j}n{)}^{-s}\mathrm{\Gamma }(s)\mathrm{d}s,\end{array}

and

\begin{array}{r}-\frac{L^{\prime }}{L}(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\mathrm{\Lambda }(n)\chi (n)n^{-s},(\mathrm{\Re }s>1).\end{array}

Therefore, when \chi \ne {\chi }_0,

\begin{array}{r}{W}_j(\chi ,\eta )=-\frac{1}{2\pi \mathrm{i}}{\int }_{(\sigma )}{z}_j^{-s}\frac{L^{\prime }}{L}(s,\chi )\mathrm{\Gamma }(s)\mathrm{d}s.\end{array}

According to the Remainder Theorem in complex analysis and Lemma 2 in [2], assuming GRH, we have

(16)\begin{array}{r}{W}_j(\chi ,\eta )=-\sum _{\rho }{z}_j^{-\rho }\mathrm{\Gamma }(\rho )+O(\log b\log q),\end{array}

where \rho is over all the non-trivial zeros of L(s,\chi ). According to Lemma 6 of [2],

\begin{array}{r}\sum _{\rho }{z}_j^{-\rho }\mathrm{\Gamma }(\rho )\ll b^{\frac{1}{2}}\log (q+1)+b|\eta {|}^{\frac{1}{2}}\log (qb).\end{array}

Hence we obtain the following lemma.

Lemma 4　 Assume the GRH, and set , \chi \mathrm{mod} q and \eta \in {\xi }_{q,a}. Then we have

(17)\begin{array}{r}{S}_{j,2}\ll ((bq{)}^{\frac{1}{2}}+b(q|\eta |{)}^{\frac{1}{2}})\log (qb).\end{array}

In particular, when Q=b^{1/2}/2, by Lemma 4, we obtain

\begin{array}{rl}{S}^{\ast }(Q):=& \underset{1⩽q⩽Q}{max}\underset{\genfrac{}{}{0ex}{}{1⩽a⩽q}{(a,q)=1}}{max}\underset{\eta \in {\xi }_{q,a}}{max}|S_{j,2}+S_{j,3}|\\ \ll & \left(\sqrt{bQ}+\frac{b}{\sqrt{Q}}\right)\log (Qb)\ll b^{3/4}\log b.\end{array}

In the following, we make mean estimation to S_{j,1} and S_{j,2}+S_{j,3}. We know that

\begin{array}{r}{S}_{j,1}=\frac{C_q(a_j)}{\varphi (q)}{V}_j(\eta ),\end{array}

where C_q( a_j)\ll (q ,a_j), V_j(\eta )⩽z_j^{-1}. Therefore

(18)\begin{array}{r}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{j,1}{|}^2\mathrm{d}\eta \ll \sum _{q=1}^Q\frac{(q,a_j{)}^2}{\varphi (q)}{\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}\frac{1}{|z_j{|}^2}\mathrm{d}\eta \ll B^{2}b\log Q,\end{array}

where B=\underset{1⩽i⩽k}{max}\{a_i\}. Further, by the Cauchy-Schwarz inequality, we have

(19)\begin{array}{rl}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{j,2}+S_{j,3}{|}^2\mathrm{d}\eta & =\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_j(\alpha )-S_{j,1}{|}^2\mathrm{d}\eta \\ & \ll {\int }_{\frac{1}{Q+1}}^{1+\frac{1}{Q+1}}|S_j(\alpha ){|}^2\mathrm{d}\eta +\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{j,1}{|}^2\mathrm{d}\eta .\end{array}

Putting inequality (18) and

\begin{array}{r}{\int }_{\frac{1}{Q+1}}^{1+\frac{1}{Q+1}}|S_j(\alpha ){|}^2\mathrm{d}\eta \ll \frac{b}{a_j}\log b\end{array}

into the right-hand side of (19), we have

(20)\begin{array}{r}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{j,2}+S_{j,3}{|}^2\mathrm{d}\eta \ll B^{2}b\log b.\end{array}

In addition, when r⩾3, we also have

(21)\begin{array}{r}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{j,1}{|}^r\mathrm{d}\eta \ll \sum _{q=1}^Q\frac{(q,a_j{)}^r}{\varphi (q{)}^{r-1}}{\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}\frac{1}{|min\{b^{-1},\eta \}{|}^r}\mathrm{d}\eta \ll B^r{b}^{r-1}.\end{array}

Now, we divide S_3 into k-1 parts according to m(0⩽m⩽k-2 ), which are

\begin{array}{r}{S}_{3,0}=\prod _{j=1}^k(S_{j,2}+S_{j,3}),S_{3,m}=\prod _{i=1}^m{S}_{i,1}\prod _{j=m+1}^k(S_{j,2}+S_{j,3}),(1⩽m⩽k-2).\end{array}

And correspondingly, M_{k,3}(b) is divided into k-1 parts, which are

\begin{array}{r}{M}_{3,m}=\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{q}e\left(-b\frac{a}{q}\right){\int }_{{\xi }_{q,a}}{S}_{3,m}e(-b\eta )\mathrm{d}\eta ,(0⩽m⩽k-2).\end{array}

We further apply inequalities (18), (20) and (21) to estimate M_{3,m}(0⩽m⩽k-2 ) as follows:

\begin{array}{rl}{M}_{3,0}& \ll (S^{\ast }(Q){)}^{k-2}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{j,2}+S_{j,3}{|}^2\mathrm{d}\eta \\ & \ll b^{\frac{3}{4}(k-2)}(\log b{)}^{k-2}{B}^{2}b\log b\ll B^2{b}^{k-\frac{7}{4}}(\log b{)}^{k-1};\end{array}

\begin{array}{rl}{M}_{3,1}& \ll (S^{\ast }(Q){)}^{k-2}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{i,1}||S_{j,2}+S_{j,3}|\mathrm{d}\eta \\ & \ll B^2{b}^{k-\frac{7}{4}}(\log b{)}^{k-1};\end{array}

M_{3,2}\ll (S^{\ast }(Q ){)}^{k-2}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\int }_{{\xi }_{q,a}}|S_{i,1}{|}^2\mathrm{d}\eta \ll B^2{b}^{k-\frac{7}{4}}(\log b {)}^{k-1}.

To estimate M_{3,r}(3 ⩽r ⩽k -2), we need the following lemma.

Lemma 5　 Assume the GRH, and let c>1. Then

\begin{array}{r}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^{\mathrm{\infty }}\frac{1}{\varphi (q{)}^c}{\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}|S_{i,1}{|}^2|S_{j,2}+S_{j,3}{|}^{2}d\eta \ll B^2{b}^2(\log b{)}^2,\end{array}

where B= \underset{1⩽i⩽k}{max}\{|a_i|\}.

In fact, by Lemma 4, S_{j,3}\ll \log ( qb) and V_i( \eta )\ll \frac{1}{min\{\frac{1}{b},\eta \}}, we can obtain the summation in Lemma 5 is

\begin{array}{rl}& \ll \sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q\frac{B^2}{\varphi (q{)}^{c+2}}{\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}((bq{)}^{\frac{1}{2}}+b(q|\eta |{)}^{\frac{1}{2}}{)}^2\frac{\mathrm{d}\eta }{|min\{{\displaystyle \frac{1}{b}},\eta \}{|}^2}\\ & \ll B^2\sum _{q=1}^Q\frac{1}{\varphi (q{)}^{c+1}}\left({\int }_0^{\frac{1}{b}}bq\cdot b^2\mathrm{d}\eta +{\int }_{\frac{1}{b}}^{\frac{1}{qQ}}{b}^{2}q\eta \cdot \frac{1}{{\eta }^2}\mathrm{d}\eta \right)\\ & \ll B^2{b}^2\sum _{q=1}^Q\frac{1}{\varphi (q{)}^c}{\int }_{\frac{1}{b}}^{\frac{1}{qQ}}\frac{1}{\eta }\mathrm{d}\eta \\ & \ll B^2{b}^2(\log b{)}^2.\end{array}

In this way, by Lemma 5, for 3⩽r⩽k-2, we have

\begin{array}{rl}{M}_{3,r}& \ll (S^8(Q){)}^{k-r-2}\sum _{q=1}^Q\sum _{\genfrac{}{}{0ex}{}{a=1}{(a,q)=1}}^q{\frac{1}{\varphi (q)}}^{r-2}{\int }_{-\frac{1}{qQ}}^{\frac{1}{qQ}}|S_{i,1}{|}^2|S_{j,2}+S_{j,3}{|}^2\mathrm{d}\eta \\ & \ll b^{\frac{3}{4}(k-r-2)}(\log b{)}^{k-r-2}\cdot B^2{b}^2(\log b{)}^2\\ & \ll B^2{b}^{k-\frac{7}{4}}(\log b{)}^{k-r}.\end{array}

Hence, we obtain that when k⩾5, assuming the GRH,

(22)\begin{array}{r}{M}_{k,3}(b)\ll B^2{b}^{k-\frac{7}{4}}(\log b{)}^{k-1}.\end{array}

In the end, putting inequality (22), Lemma 1 and Lemma 3 into equation (6), we finish the proof of Theorem 1.

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