Some Identities Involving Certain Hardy Sums and General Kloosterman Sums

Abstract

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums $K(r,l,\lambda ;p)$ and certain Hardy sums $S_1(h,q)$ ${\displaystyle \sum _{m=1}^{p-1}}$${\displaystyle \sum _{s=1}^{p-1}}$$K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p),$ ${\displaystyle \sum _{m=1}^{p-1}}$${\displaystyle \sum _{s=1}^{p-1}}$${|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p).$ Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character $\lambda$ modulo p, while we consider all the cases.

1. Introduction

Let k be a positive integer and h be arbitrary integer with $(h,k)=1$, the classical Dedekind sums $S(h,k)$ are defined as

$$S(h,k)=\sum _{a=1}^k\left(\left(\frac{a}{k}\right)\right)\left(\left(\frac{ah}{k}\right)\right),$$

where

$$\left(\left(x\right)\right)=\left\{\begin{array}{cc}x-[x]-\frac{1}{2},\hfill & \mathrm{if}x\mathrm{is}\mathrm{not}\mathrm{an}\mathrm{integer};\hfill \\ 0,\hfill & \mathrm{if}x\mathrm{is}\mathrm{an}\mathrm{integer}.\hfill \end{array}\right.$$

$S(h,k)$ also can be written as

$$S(h,k)=-\frac{1}{4k}\sum _{a=1}^{k-1}cot\frac{\pi a}{k}cot\frac{\pi ah}{k},$$

which belong to the family of “cotangent sums”.

These sums have a wide range of applications and in some cases relations to some major open problems, such as Riemann Hypothesis. Refs. [1,2,3,4] introduced the cotangent sums

$$C\left(\frac{h}{k}\right)=\sum _{a=1}^{k-1}\frac{a}{k}cot\frac{\pi ah}{k},$$

and the Vasyunin sums

$$V\left(\frac{h}{k}\right)=\sum _{a=1}^{k-1}\left\{\frac{ah}{k}\right\}cot\frac{\pi ah}{k}=C\left(\frac{\overline{h}}{k}\right),$$

where $\{u\}=u-\lfloor u\rfloor ,h\overline{h}\equiv 1modk.$ Actually the Vasyunin sums are associated to the study of the Riemann Hypothesis through the following equation (see [1,2,3,4]):

$$\begin{array}{c}\frac{1}{2\pi \sqrt{hk}}{\int }_{-\infty }^{\infty }{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2{\left(\frac{h}{k}\right)}^{it}\frac{dt}{\frac{1}{4}+t^2}\hfill \\ =\frac{log2\pi -\gamma }{2}\left(\frac{1}{h}+\frac{1}{k}\right)+\frac{k-h}{2hk}log\frac{h}{k}-\frac{\pi }{2hk}\left(V\left(\frac{h}{k}\right)+V\left(\frac{k}{h}\right)\right).\hfill \end{array}$$

So the cotangent sums arise in connection with the Nyman-Beurling approach to the Riemann Hypothesis.

Dedekind sums also play an important role in the transformation theory of the Dedekind $\eta$ function. Dedekind sums have many interesting properties. For example, L. Carlitz [5] obtained the reciprocity theorem of $S(h,k)$ as

$$S(h,k)+S(k,h)=-\frac{1}{4}+\frac{1}{12}\left(\frac{h}{k}+\frac{k}{h}+\frac{1}{hk}\right).$$

W. P. Zhang [6] established the relationship between Dedekind sums and Dirichlet L-function:

$$S(a,k)=\frac{1}{{\pi }^{2}k}\sum _{d|k}\frac{d^2}{\phi (d)}\underset{\chi (-1)=-1}{\sum _{\chi modd}}\chi (a){|L(1,\chi )|}^2,$$

(1)

where $k>2$ is an integer, a is an integer with $(a,k)=1$, $\phi (d)$ denotes the Euler function and $L(1,\chi )$ denotes the Dirichlet L-function corresponding to character $\chi$ modulo d.

Some scholars also studied Dedekind type sums and obtained interesting results. For example, by considering Dedekind type DC(Daehee-Changhee) sums

$$T_p(h,k)=2\sum _{u=1}^{k-1}{(-1)}^{u-1}\frac{u}{k}{\overline{E}}_p\left(\frac{hu}{k}\right),(h\in Z_{+}),$$

where ${\overline{E}}_p(x)$ are the p-th Euler functions, T. Kim [7] proved reciprocity law:

$$\begin{array}{c}{k}^p{T}_p(h,k)+h^p{T}_p(k,h)\hfill \\ =2\underset{u-[\frac{hu}{k}]\equiv 1mod2}{\sum _{u=0}^{k-1}}{\left(kh\left(E+\frac{u}{k}\right)+k\left(E+h-\left[\frac{hu}{k}\right]\right)\right)}^p+{(hE+kE)}^p+(p+2)E_p,\hfill \end{array}$$

where $(h,k)=1$, $[x]$ is the largest integer $\le x$, $E_n$ are the n-th Euler numbers. Later T. Kim [8] defined p-adic Dedekind-type DC sums as follows:

$$S_{p,q}(s:h,k:q^k)=\sum _{M=1}^{k-1}\left(\frac{1-q^M}{1-q}\right){(-1)}^{M-1}{T}_q(s,hM,k:q^k),$$

where $(h,k)=(p,k)=1,s\in Z_p$, and $T_q(m,a,N:q^N)$ is a continuous p-adic extension of ${(\frac{1-q^N}{1-q})}^m{E}_{m,q^N}(\frac{a}{N})$, then he got a continuous function $S_{p,q}(m:h,k:q^k)$ on $Z_p$, which satisfies

$$S_{p,q}(m:h,k:q^k)={\left(\frac{1-q^k}{1-q}\right)}^{m+1}{S}_{m,q}(h,k:q^k)-{\left(\frac{1-q^k}{1-q}\right)}^{m+1}{\left(\frac{1-q^{kp}}{1-q^k}\right)}^m{S}_{m,q}({(p^{-1}h)}_k,k:q^{pk}),$$

where

$$S_{m,q}(h,k:q^l)=\sum _{M=1}^{k-1}{(-1)}^{M-1}\left(\frac{1-q^M}{1-q^k}\right){\int }_{Z_p}{q}^{-lx}{\left(\frac{1-q^{l(x+\{\frac{hM}{k}\})}}{1-q^l}\right)}^{m}d{\mu }_{q^l}(x).$$

The other sums analogous to Dedekind sums are defined as

$$S_1(h,k)=\sum _{j=1}^{k-1}{(-1)}^{j+1+[\frac{hj}{k}]},$$

where h and k are integers with $k>0$. The sums $S_1(h,k)$ are sometimes called Hardy sums. Some authors studied the properties of $S_1(h,k)$ and related sums, and obtained some interesting results, see [9,10,11,12]. A relation between certain Hardy sums $S_1(h,k)$ and classical Dedekind sums $S(h,k)$ can be obtained in [12] that if $(h,k)=1$, then

$$S_1(h,k)=-8S(h+k,2k)+4S(h,k).$$

Other scholars showed their interests to the hybrid mean value involving Hardy sums and other famous sums, see [13,14,15]. For example, W. P. Zhang [15] studied the hybrid mean value involving certain Hardy sums $S_1(h,k)$ and Kloosterman sums

$$K(r,q)=\underset{a=1}{\overset{q}{{\sum }^{\prime }}}e\left(\frac{ra+\overline{a}}{q}\right),$$

where ${\displaystyle \underset{a=1}{\overset{q}{{\sum }^{\prime }}}}$ denotes the summation over all a with $(a,q)=1,$$e(y)=e^{2\pi iy}$, and obtained some exact computational formulas for

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,p)K(s,p)S_1(2m\overline{s},p),$$

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,p)|}^2{|K(s,p)|}^2{S}_1(2m\overline{s},p).$$

Actually the transforming formula (1) and

$$S_1(2h,p)=-20S(2h,p)+8S(4h,p)+8S(h,p)$$

are used, where p is an odd prime and $0<h<p$. Therefore,

$$\begin{array}{ccc}\hfill S_1(2m\overline{s},p)& =& -\frac{20p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2m\overline{s}){\left|L(1,\chi )\right|}^2\hfill \\ & & +\frac{8p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4m\overline{s}){\left|L(1,\chi )\right|}^2\hfill \\ & & +\frac{8p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (m\overline{s}){\left|L(1,\chi )\right|}^2.\hfill \end{array}$$

(2)

However, as far as we know, it seems that nobody has yet studied the hybrid mean value involving certain Hardy sums $S_1(h,k)$ and general Kloosterman sums $K(r,l,\lambda ;p)$, which are defined as follows:

$$K(r,l,\lambda ;q)=\underset{a=1}{\overset{q}{{\sum }^{\prime }}}\lambda (a)e\left(\frac{ra+l\overline{a}}{q}\right),$$

where $\lambda$ is a Dirichlet character modulo q. If $\lambda ={\lambda }_0$ is the principal character modulo q and $l=1$, then $K(r,1,{\lambda }_0;q)=K(r,q).$

Inspired by [15], we will study the hybrid mean value

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p),$$

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p).$$

Since they are more comprehensive than those in [15], we turn to some novel methods and finally obtain several explicit formulas. Our results not only cover those of [15], but also contain something quite new.

Theorem 1.

Let p be an odd prime. Then for any character λ $mod$ p and any integers n, t with $(n,p)=(t,p)=1$, if $p\equiv 1mod4$, we have the identity

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p)=0.$$

If $p\equiv 3mod4$, we have

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p)=\left\{\begin{array}{cc}2p,\hfill & \mathrm{if}\overline{\lambda }\chi ={\chi }_0;\hfill \\ 2p^2,\hfill & \mathrm{if}\overline{\lambda }\chi \ne {\chi }_0,\hfill \end{array}\right.$$

where χ is any odd character modulo p and ${\chi }_0$ is the principal character modulo p.

Note: The emergences of $\chi$ in Theorem 1 are due to the application of (2) in the proof.

Theorem 2.

Let p be an odd prime. Then for any character λ $mod$ p and any integers n, t with $(n,p)=(t,p)=1$, if $p\equiv 1mod4$, we have the identity

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{\left|K(m,n,\lambda ;p)\right|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)=0.$$

If $p\equiv 3mod8$, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ =\left\{\begin{array}{cc}2p^3-36p^2{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ 2p^2-36p{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[p^2-p-54h_p^2+1+2(p-1-18h_p^2)Re\tau ({\overline{\chi }}^2)],}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[2p^2-2p-18p{h}_p^2+1-2(p-1-18h_p^2)Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )}\hfill & \\ +36h_p^{2}Re\tau (\overline{\lambda }\chi )],\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.\hfill \end{array}$$

If $p\equiv 7mod8$, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ =\left\{\begin{array}{cc}2p^3+4p^2{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ 2p^2+4p{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[p^2-p+6h_p^2+1+2(p-1+2h_p^2)Re\tau ({\overline{\chi }}^2)],}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[2p^2-2p+2p{h}_p^2+1-2(p-1+2h_p^2)Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )}\hfill & \\ +4h_p^{2}Re\tau (\overline{\lambda }\chi )],\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0,\hfill \end{array}\right.\hfill \end{array}$$

where $h_p$ denotes the class number of the quadratic field $Q(\sqrt{-p})$, χ is any odd character modulo p, and ${\displaystyle \tau (\chi )=\sum _{a=1}^{p-1}\chi (a)e\left(\frac{a}{p}\right)}$ denotes the classical Gauss sums.

Note: We know that $|\tau (\chi )|=\sqrt{p}$ if $\chi$ is a primitive character modulo p. For the case of $\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0$, taking $\sqrt{p}$ as the upper bound estimate of $Re\tau ({\overline{\chi }}^2)$, we can get

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)=\left\{\begin{array}{cc}2p^4+O(p^{\frac{7}{2}}),\hfill & \mathrm{if}p\equiv 3mod8;\hfill \\ 2p^4+O(p^{\frac{7}{2}}),\hfill & \mathrm{if}p\equiv 7mod8.\hfill \end{array}\right.\hfill \end{array}$$

Let $n=t=1,$ $\lambda ={\lambda }_0$ in Theorems, we may immediately obtain Theorems 1 and 2 of [15] as the following:

Corollary 1.

Let p be an odd prime, then we have

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,p)K(s,p)S_1(2m\overline{s},p)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}p\equiv 1mod4;\hfill \\ 2p^2,\hfill & \mathrm{if}p\equiv 3mod4.\hfill \end{array}\right.$$

Corollary 2.

Let p be an odd prime, then we have

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,p)|}^2{|K(s,p)|}^2{S}_1(2m\overline{s},p)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}p\equiv 1mod4;\hfill \\ 2p^3-36p^2{h}_p^2,\hfill & \mathrm{if}p\equiv 3mod8;\hfill \\ 2p^3+4p^2{h}_p^2,\hfill & \mathrm{if}p\equiv 7mod8.\hfill \end{array}\right.$$

It should be pointed out that we only consider the prime modulus case. The question of whether there exist some exact computational formulas for the general modulus q remains open.

2. Some Lemmas

In this section, we will give several simple lemmas, which are necessary in the proof of our theorems. Hereinafter, we shall use some knowledge of elementary number theory, the orthogonality relation of character sum and the properties of Gauss sums, which all can be found in [16], here we only list a few. For example,

$${\displaystyle \sum _{a=1}^{p-1}\chi (a)e\left(\frac{ua}{p}\right)=\tau (\chi )\overline{\chi }(u),}$$

if $(u,p)=1$ or $\chi$ is a primitive character modulo p. If $\chi$ is a primitive character modulo p, then $|\tau (\chi )|=\sqrt{p}$.

Now we have the following lemmas:

Lemma 1.

Let p be an odd prime, n be any integer with $(n,p)=1$. Then for any non-principal character χ $mod$ p and any character λ$mod$ p, we have

$$\left|\sum _{m=1}^{p-1}\chi (m){|K(m,n,\lambda ;p)|}^2\right|=\left\{\begin{array}{cc}p|\tau ({\overline{\chi }}^2)|,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ p^{\frac{1}{2}}|\tau ({\overline{\chi }}^2)|,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ p|\tau ({\overline{\chi }}^2)+(p-1)|,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ p|-\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )+(p-1))|,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.$$

Proof. 

From the properties of Gauss sums and the reduced residue system modulo p, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\chi (m){|K(m,n,\lambda ;p)|}^2\hfill \\ =\sum _{m=1}^{p-1}\chi (m){\left|\sum _{a=1}^{p-1}\lambda (a)e\left(\frac{ma+n\overline{a}}{p}\right)\right|}^2\hfill \\ =\sum _{a=1}^{p-1}\sum _{b=1}^{p-1}\lambda (a\overline{b})\sum _{m=1}^{p-1}\chi (m)e\left(\frac{m(a-b)+n(\overline{a}-\overline{b})}{p}\right)\hfill \\ =\sum _{a=1}^{p-1}\lambda (a)\sum _{b=1}^{p-1}\sum _{m=1}^{p-1}\chi (m)e\left(\frac{mb(a-1)+n\overline{b}(\overline{a}-1)}{p}\right)\hfill \\ =\sum _{a=1}^{p-1}\lambda (a)\sum _{b=1}^{p-1}e\left(\frac{n\overline{b}(\overline{a}-1)}{p}\right)\sum _{m=1}^{p-1}\chi (m)e\left(\frac{mb(a-1)}{p}\right)\hfill \\ =\tau (\chi )\sum _{a=1}^{p-1}\lambda (a)\overline{\chi }(a-1)\sum _{b=1}^{p-1}\chi (b)e\left(\frac{nb(\overline{a}-1)}{p}\right)\hfill \\ ={\tau }^2(\chi )\overline{\chi }(n)\sum _{a=1}^{p-1}\lambda (a)\overline{\chi }(a-1)\overline{\chi }(\overline{a}-1)\hfill \\ ={\tau }^2(\chi )\overline{\chi }(n)\sum _{a=1}^{p-1}\lambda (a)\overline{\chi }(2-a-\overline{a})\hfill \\ ={\tau }^2(\chi )\overline{\chi }(n)\sum _{a=1}^{p-1}\lambda (a)\overline{\chi }(\overline{a})\overline{\chi }(2a-a^2-1)\hfill \\ ={\tau }^2(\chi )\overline{\chi }(-n)\sum _{a=1}^{p-1}\lambda \chi (a)\overline{\chi }({(a-1)}^2)\hfill \\ ={\tau }^2(\chi )\overline{\chi }(-n)\sum _{a=1}^{p-2}\lambda \chi (a+1){\overline{\chi }}^2(a).\hfill \end{array}$$

(3)

Using the properties of Gauss sums and the periodicity of Dirichlet character, we have

$$\begin{array}{c}\sum _{a=1}^{p-2}\lambda \chi (a+1){\overline{\chi }}^2(a)\hfill \\ =\frac{1}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda \chi }(b)\sum _{a=1}^{p-2}{\overline{\chi }}^2(a)e\left(\frac{b(a+1)}{p}\right)\hfill \\ =\frac{1}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda \chi }(b)e\left(\frac{b}{p}\right)\sum _{a=1}^{p-2}{\overline{\chi }}^2(a)e\left(\frac{ab}{p}\right)\hfill \\ =\frac{1}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda \chi }(b)e\left(\frac{b}{p}\right)\left(\sum _{a=1}^{p-1}{\overline{\chi }}^2(a)e\left(\frac{ab}{p}\right)-{\overline{\chi }}^2(p-1)e\left(\frac{pb-b}{p}\right)\right)\hfill \\ =\frac{1}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda \chi }(b)e\left(\frac{b}{p}\right)\left(\sum _{a=1}^{p-1}{\overline{\chi }}^2(a)e\left(\frac{ab}{p}\right)-e\left(\frac{-b}{p}\right)\right)\hfill \\ =\frac{\tau ({\overline{\chi }}^2)}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda }\chi (b)e\left(\frac{b}{p}\right)-\frac{1}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda \chi }(b)\hfill \\ =\frac{\tau (\overline{\lambda }\chi )\tau ({\overline{\chi }}^2)}{\tau (\overline{\lambda \chi })}-\frac{1}{\tau (\overline{\lambda \chi })}\sum _{b=1}^{p-1}\overline{\lambda \chi }(b).\hfill \end{array}$$

Applying the orthogonality relation for character modulo p, it is clear that

$$\sum _{b=1}^{p-1}\overline{\lambda \chi }(b)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0;\hfill \\ p-1,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0.\hfill \end{array}\right.$$

And note that $\tau ({\chi }_0)=-1$, so we can get

$$\begin{array}{c}\sum _{a=1}^{p-2}\lambda \chi (a+1){\overline{\chi }}^2(a)\hfill \\ =\left\{\begin{array}{cc}{\displaystyle \frac{\tau (\overline{\lambda }\chi )\tau ({\overline{\chi }}^2)}{\tau (\overline{\lambda \chi })},}\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0;\hfill \\ {\displaystyle -\tau (\overline{\lambda }\chi )\tau ({\overline{\chi }}^2)+(p-1),}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0.\hfill \end{array}\right.\hfill \\ =\left\{\begin{array}{cc}{\displaystyle \frac{\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )}{\tau (\overline{\lambda \chi })},}\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ {\displaystyle \frac{-\tau ({\overline{\chi }}^2)}{\tau (\overline{\lambda \chi })},}\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle \tau ({\overline{\chi }}^2)+(p-1),}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle -\tau (\overline{\lambda }\chi )\tau ({\overline{\chi }}^2)+(p-1),}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.\hfill \end{array}$$

(4)

Note that $|\overline{\chi }(-n)|=1$ and $|\tau (\overline{\lambda \chi })|=|\tau (\overline{\lambda }\chi )|=|\tau (\chi )|=\sqrt{p}$ if $\overline{\lambda \chi }$ and $\overline{\lambda }\chi$ are all non-principal characters modulo p. Thus, from (3) and (4) we may immediately deduce that

$$\begin{array}{c}\left|\sum _{m=1}^{p-1}\chi (m){|K(m,n,\lambda ;p)|}^2\right|\hfill \\ =\left\{\begin{array}{cc}p\left|\tau ({\overline{\chi }}^2)\right|,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ p^{\frac{1}{2}}\left|\tau ({\overline{\chi }}^2)\right|,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ p\left|(\tau ({\overline{\chi }}^2)+(p-1))\right|,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ p\left|(-\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )+(p-1))\right|,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.\hfill \end{array}$$

This proves Lemma 1. □

Lemma 2.

Let p be an odd prime, n be any integer with $(n,p)=1$. Then for any non-principal character χ $mod$ p and any character λ $mod$ p, we have

$$\left|\sum _{m=1}^{p-1}\chi (m)K(m,n,\lambda ;p)\right|=\left\{\begin{array}{cc}{p}^{\frac{1}{2}},\hfill & \mathrm{if}\overline{\lambda }\chi ={\chi }_0;\hfill \\ p,\hfill & \mathrm{if}\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.$$

Proof. 

From the definition of Kloosterman sums and the properties of Gauss sums we have

$$\begin{array}{c}\left|\sum _{m=1}^{p-1}\chi (m)K(m,n,\lambda ;p)\right|\hfill \\ =\left|\sum _{m=1}^{p-1}\chi (m)\sum _{a=1}^{p-1}\lambda (a)e\left(\frac{ma+n\overline{a}}{p}\right)\right|\hfill \\ =\left|\sum _{a=1}^{p-1}\lambda (a)e\left(\frac{n\overline{a}}{p}\right)\sum _{m=1}^{p-1}\chi (m)e\left(\frac{ma}{p}\right)\right|\hfill \\ =\left|\tau (\chi )\sum _{a=1}^{p-1}\lambda (a)\overline{\chi }(a)e\left(\frac{n\overline{a}}{p}\right)\right|\hfill \\ =\left|\tau (\chi )\sum _{a=1}^{p-1}\overline{\lambda }\chi (a)e\left(\frac{na}{p}\right)\right|\hfill \\ =\left|\tau (\chi )\tau (\overline{\lambda }\chi )\lambda \overline{\chi }(n)\right|.\hfill \end{array}$$

Note that $|\tau (\chi )|=\sqrt{p},\left|\lambda \overline{\chi }(n)\right|=1,\tau ({\chi }_0)=-1,$ then we can immediately deduce that

$$\begin{array}{c}\left|\sum _{m=1}^{p-1}\chi (m)K(m,n,\lambda ;p)\right|=\left\{\begin{array}{cc}{p}^{\frac{1}{2}},\hfill & \mathrm{if}\overline{\lambda }\chi ={\chi }_0;\hfill \\ p,\hfill & \mathrm{if}\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.\hfill \end{array}$$

This proves Lemma 2. □

Lemma 3.

Let p be an odd prime, then we have

$${\underset{\chi (-1)=-1}{\sum _{\chi modp}}|L(1,\chi )|}^2=\frac{{\pi }^2}{12}\frac{{(p-1)}^2(p-2)}{p^2},$$

$${\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2=\frac{{\pi }^2}{24}\frac{{(p-1)}^2(p-5)}{p^2},$$

$${\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4)|L(1,\chi )|}^2=\left\{\begin{array}{cc}{\displaystyle \frac{{\pi }^2}{48}\frac{{(p-1)}^2(p-17)}{p^2},}\hfill & p\equiv 1mod4;\hfill \\ {\displaystyle \frac{{\pi }^2}{48}\frac{(p-1)(p^2-6p+17)}{p^2},}\hfill & p\equiv 3mod4.\hfill \end{array}\right.$$

Proof. 

See Lemma 5 of [15]. □

3. Proof of the Theorems

In this section, we shall use the above lemmas to complete the proof of Theorems. First we prove Theorem 1 From the identity (2), we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p)\hfill \\ =-\frac{20p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2){\left|\sum _{m=1}^{p-1}\chi (m)K(m,n,\lambda ;p)\right|}^2{\left|L(1,\chi )\right|}^2\hfill \\ +\frac{8p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){\left|\sum _{m=1}^{p-1}\chi (m)K(m,n,\lambda ;p)\right|}^2{\left|L(1,\chi )\right|}^2\hfill \\ +\frac{8p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}{\left|\sum _{m=1}^{p-1}\chi (m)K(m,n,\lambda ;p)\right|}^2{\left|L(1,\chi )\right|}^2.\hfill \end{array}$$

Firstly, we consider the case of $p\equiv 1mod4$. If $\overline{\lambda }\chi ={\chi }_0$, from Lemmas 2 and 3, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p)\hfill \\ {\displaystyle =-\frac{20p^2}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2+\frac{8p^2}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){|L(1,\chi )|}^2}\hfill \\ {\displaystyle +\frac{8p^2}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}{|L(1,\chi )|}^2}\hfill \\ {\displaystyle =-\frac{5}{6}(p-1)(p-5)+\frac{2}{3}(p-1)(p-2)+\frac{1}{6}(p-1)(p-17)}\hfill \\ =0.\hfill \end{array}$$

If $\overline{\lambda }\chi \ne {\chi }_0$, then from Lemmas 2 and 3, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p)\hfill \\ {\displaystyle =-\frac{20p^3}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2+\frac{8p^3}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){|L(1,\chi )|}^2}\hfill \\ {\displaystyle +\frac{8p^3}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}{|L(1,\chi )|}^2}\hfill \\ {\displaystyle =-\frac{5}{6}p(p-1)(p-5)+\frac{2}{3}p(p-1)(p-2)+\frac{1}{6}p(p-1)(p-17)}\hfill \\ =0.\hfill \end{array}$$

For the case of $p\equiv 3mod4$, similarly we have

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}K(m,n,\lambda ;p)K(s,t,\lambda ;p)S_1(2m\overline{s},p)=\left\{\begin{array}{cc}2p,\hfill & \mathrm{if}\overline{\lambda }\chi ={\chi }_0;\hfill \\ 2p^2,\hfill & \mathrm{if}\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.$$

This completes the proof of Theorem 1.

Now we prove Theorem 2. From the identity (2), it is clear that

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ =-\frac{20p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2){\left|\sum _{m=1}^{p-1}\chi (m){|K(m,n,\lambda ;p)|}^2\right|}^2{|L(1,\chi )|}^2\hfill \\ +\frac{8p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){\left|\sum _{m=1}^{p-1}\chi (m){|K(m,n,\lambda ;p)|}^2\right|}^2{|L(1,\chi )|}^2\hfill \\ +\frac{8p}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}{\left|\sum _{m=1}^{p-1}\chi (m){|K(m,n,\lambda ;p)|}^2\right|}^2{|L(1,\chi )|}^2.\hfill \end{array}$$

Firstly, we consider the case of $p\equiv 1mod4$. Note that $|\tau ({\overline{\chi }}^2)|=\sqrt{p}$. If $\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0$, from Lemmas 1 and 3, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ {\displaystyle =-\frac{20p^4}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2+\frac{8p^4}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){|L(1,\chi )|}^2}\hfill \\ {\displaystyle +\frac{8p^4}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}{|L(1,\chi )|}^2}\hfill \\ {\displaystyle =-\frac{5}{6}{p}^2(p-1)(p-5)+\frac{1}{6}{p}^2(p-1)(p-17)+\frac{2}{3}{p}^2(p-1)(p-2)}\hfill \\ =0.\hfill \end{array}$$

If $\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0$, from Lemmas 1 and 3, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ {\displaystyle =-\frac{20p^3}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2+\frac{8p^3}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){|L(1,\chi )|}^2}\hfill \\ {\displaystyle +\frac{8p^3}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}{|L(1,\chi )|}^2}\hfill \\ {\displaystyle =-\frac{5}{6}p(p-1)(p-5)+\frac{1}{6}p(p-1)(p-17)+\frac{2}{3}p(p-1)(p-2)}\hfill \\ =0.\hfill \end{array}$$

Similarly, we have

$$\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ 0,\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.$$

Now, we consider the case of $p\equiv 3mod4$. In this case, we note that

$${\displaystyle \left(\frac{-1}{p}\right)={\chi }_2(-1)=-1,L(1,{\chi }_2)=\frac{\pi h_p}{\sqrt{p}},\tau ({\overline{{\chi }_2}}^2)=-1,\left(\frac{4}{p}\right)=\left(\frac{2^2}{p}\right)=1.}$$

So, if $\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0$, from Lemmas 1 and 3, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ {\displaystyle =-\frac{20p^4}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2+\frac{8p^4}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){|L(1,\chi )|}^2}\hfill \\ {\displaystyle +\frac{8p^4}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}|L(1,\chi )|}^2+\frac{20p^4}{{\pi }^2(p-1)}{\chi }_2(2){|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle -\frac{20p^3}{{\pi }^2(p-1)}{\chi }_2(2)|L(1,{\chi }_2){|}^2-\frac{8p^4}{{\pi }^2(p-1)}{\chi }_2(4)|L(1,{\chi }_2){|}^2+\frac{8p^3}{{\pi }^2(p-1)}{\chi }_2(4){|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle -\frac{8p^4}{{\pi }^2(p-1)}|L(1,{\chi }_2){|}^2+\frac{8p^3}{{\pi }^2(p-1)}{|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle =-\frac{5}{6}{p}^2(p-1)(p-5)+\frac{1}{6}{p}^2(p^2-6p+17)+\frac{2}{3}{p}^2(p-1)(p-2)}\hfill \\ {\displaystyle +\frac{20p^3}{{\pi }^2}{\chi }_2(2)|L(1,{\chi }_2){|}^2-\frac{16p^3}{{\pi }^2}{|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle =2p^3-16p^2{h}_p^2+20p^2{h}_p^2\left(\frac{2}{p}\right).}\hfill \end{array}$$

If $\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0$, from Lemmas 1 and 3, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ {\displaystyle =-\frac{20p^3}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (2)|L(1,\chi )|}^2+\frac{8p^3}{{\pi }^2(p-1)}\underset{\chi (-1)=-1}{\sum _{\chi modp}}\chi (4){|L(1,\chi )|}^2}\hfill \\ {\displaystyle +\frac{8p^3}{{\pi }^2(p-1)}{\underset{\chi (-1)=-1}{\sum _{\chi modp}}|L(1,\chi )|}^2+\frac{20p^3}{{\pi }^2(p-1)}{\chi }_2(2){|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle -\frac{20p^2}{{\pi }^2(p-1)}{\chi }_2(2)|L(1,{\chi }_2){|}^2-\frac{8p^3}{{\pi }^2(p-1)}{\chi }_2(4)|L(1,{\chi }_2){|}^2+\frac{8p^2}{{\pi }^2(p-1)}{\chi }_2(4){|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle -\frac{8p^3}{{\pi }^2(p-1)}|L(1,{\chi }_2){|}^2+\frac{8p^2}{{\pi }^2(p-1)}{|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle =-\frac{5}{6}p(p-1)(p-5)+\frac{1}{6}p(p^2-6p+17)+\frac{2}{3}p(p-1)(p-2)}\hfill \\ {\displaystyle +\frac{20p^2}{{\pi }^2}{\chi }_2(2)|L(1,{\chi }_2){|}^2-\frac{16p^2}{{\pi }^2}{|L(1,{\chi }_2)|}^2}\hfill \\ {\displaystyle =2p^2-16p{h}_p^2+20p{h}_p^2\left(\frac{2}{p}\right).}\hfill \end{array}$$

Similarly, if $\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0$, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ {\displaystyle =2p^2[p+{(p-1)}^2+2(p-1)Re\tau ({\overline{\chi }}^2)]}\hfill \\ {\displaystyle +20p^2{h}_p^2(3+2Re\tau ({\overline{\chi }}^2))\left(\frac{2}{p}\right)-16p^2{h}_p^2(3+2Re\tau ({\overline{\chi }}^2)).}\hfill \end{array}$$

If $\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0$, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ {\displaystyle =2p^2[p^2+{(p-1)}^2-2(p-1)Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )]}\hfill \\ {\displaystyle +20p^2{h}_p^2(p-2(Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )+Re\tau (\overline{\lambda }\chi )))\left(\frac{2}{p}\right)}\hfill \\ {\displaystyle -16p^2{h}_p^2(p-2(Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )+Re\tau (\overline{\lambda }\chi ))).}\hfill \end{array}$$

Then if $p\equiv 3mod8$, note that ${\displaystyle \left(\frac{2}{p}\right)={(-1)}^{\frac{p^2-1}{8}}=-1}$, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ =\left\{\begin{array}{cc}2p^3-36p^2{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ 2p^2-36p{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[p^2-p-54h_p^2+1+2(p-1-18h_p^2)Re\tau ({\overline{\chi }}^2)],}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[2p^2-2p-18p{h}_p^2+1-2(p-1-18h_p^2)Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )}\hfill & \\ +36h_p^{2}Re\tau (\overline{\lambda }\chi )],\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.\hfill \end{array}$$

If $p\equiv 7mod8$, note that ${\displaystyle \left(\frac{2}{p}\right)={(-1)}^{\frac{p^2-1}{8}}=1}$, we have

$$\begin{array}{c}\sum _{m=1}^{p-1}\sum _{s=1}^{p-1}{|K(m,n,\lambda ;p)|}^2{|K(s,t,\lambda ;p)|}^2{S}_1(2m\overline{s},p)\hfill \\ =\left\{\begin{array}{cc}2p^3+4p^2{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi \ne {\chi }_0;\hfill \\ 2p^2+4p{h}_p^2,\hfill & \mathrm{if}\overline{\lambda \chi }\ne {\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[p^2-p+6h_p^2+1+2(p-1+2h_p^2)Re\tau ({\overline{\chi }}^2)],}\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi ={\chi }_0;\hfill \\ {\displaystyle 2p^2[2p^2-2p+2p{h}_p^2+1-2(p-1+2h_p^2)Re\tau ({\overline{\chi }}^2)\tau (\overline{\lambda }\chi )}\hfill & \\ +4h_p^{2}Re\tau (\overline{\lambda }\chi )],\hfill & \mathrm{if}\overline{\lambda \chi }={\chi }_0,\overline{\lambda }\chi \ne {\chi }_0.\hfill \end{array}\right.\hfill \end{array}$$

This completes the proof of Theorem 2.

4. Conclusions

In this paper, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums and certain Hardy sums. We also prove some identities in the second part by using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, which are necessary for the proof of our Theorems. We only consider the prime modulus case. The question of whether there exist some exact computational formulas for the general modulus q remains open.