New identities involving Hardy sums $S_3(h, k)$ and general Kloosterman sums

1.   Introduction and main results

If $h$ and $k$ are integers with $k > 0$, the classical Dedekind sums $S(h, k)$ are defined as

where

The various properties of $S(h, k)$ were investigated by many authors, one of which is reciprocity theorem (see Tom M. Apostol ^[1] or L. Carlitz ^[2]). That is, for all positive integers $h$ and $k$ with $(h, k) = 1$, we have the identity

Conrey et al. ^[3] studied the mean value distribution of $S(h, k)$ and deduced the important asymptotic formula

where $\begin{eqnarray*} \mathop{{\sum}'}^{k}_{h = 1}\end{eqnarray*}$ denotes the summation over all $h$ such that $(h, k) = 1$ and

Moreover, X. L. He and W. P. Zhang ^[4] gave an interesting asymptotic formula for the Dedekind sums with a weight of Hurwitz zeta-function as follows:

Other sums analogous to the Dedekind sums are the Hardy sums. Using the notation of Berndt and Goldberg ^[5], they defined

where $h$ and $k$ are integers with $k > 0$.

In 2014, H. Zhang and W. P. Zhang ^[6] obtained some beautiful identities involving $S_{1}(h, k)$ in the forms of

where $K(n, p)$ denotes the reduced form of the general Kloosterman sums attached to a Dirichlet character $\lambda$ modulo $k$ as

where $e(x) = e^{2 \pi i x}$, $\overline{a}$ denotes the solution of the congruence $x \cdot a\equiv 1 \bmod k$.

Recently, H. F. Zhang and T. P. Zhang ^[7] extended the results in ^[6] to a more general situation as

where $\overline{K(n, t, \lambda; p)}$ denotes complex conjugate of $K(n, t, \lambda; p)$.

Actually there are six forms of Hardy sums (see Berndt ^[8] and Goldberg ^[9]). A natural question is whether we can obtain similar results by replacing $S_{1}(h, k)$ with other forms of Hardy sums. Due to some technical reasons, for most of other forms of Hardy sums, the answer is no! Thanks to the important relationships among Hardy sums and Dedekind sums built by R. Sitaramachandrarao ^[10], we are lucky to find the only one $S_{3}(h, p)$ to replace, with

Our starting point relies heavily on the following in ^[10] as:

Proposition 1. Let $k$ be an odd positive integer, $h$ be an integer with $(h, k) = 1$. Then

Then applying the properties of Gauss sums and the mean square value of Dirichlet $L$-functions, we have

Theorem 1. Let $p$ be an odd prime. Then for any Dirichlet character $\lambda \bmod p$ and any integer $s$, $t$ with $(s, p) = (t, p) = 1$, we have

where $\chi$ is an odd Dirichlet character modulo $p$ and $\chi_{0}$ is the principal character modulo $p$.

Theorem 2. Let $p$ be an odd prime with $p\equiv 1 \bmod 4$. Then for any Dirichlet character $\lambda \bmod p$ and any integer $s$, $t$ with $(s, p) = (t, p) = 1$, we have

Theorem 3. Let $p$ be an odd prime with $p\equiv 3 \bmod 8$. Then for any Dirichlet character $\lambda \bmod p$ and any integer $s$, $t$ with $(s, p) = (t, p) = 1$, we have

where $h_{p}$ denotes the class number of the quadratic field $Q\left(\sqrt{-p}\right)$.

Theorem 4. Let $p$ be an odd prime with $p\equiv 7 \bmod 8$. Then for any Dirichlet character $\lambda \bmod p$ and any integer $s$, $t$ with $(s, p) = (t, p) = 1$, we have

Taking $\lambda = \lambda_{0}$, $s = t = 1$ in Theorems 1–4, we immediately deduce the following results.

Corollary 1. Let $p$ be an odd prime. Then we have the identity

Corollary 2. Let $p$ be an odd prime. Then we have

2.   Some Lemmas

To prove the Theorems, we need the following Lemmas.

Lemma 1. Let $k > 2$ be an integer. Then for any integer $a$ with $(a, k) = 1,$ we have the identity

where $L(1, \chi)$ denotes the Dirichlet $L$-function corresponding to Dirichlet character $\chi$ mod $d$.

Proof. See Lemma 2 of ^[11].

Lemma 2. Let $p$ be an odd prime, $s$ be any integer with $(s, p) = 1$. Then for any non-principal character $\chi \bmod p$ and any Dirichlet character $\lambda \bmod p$, we have

Proof. See Lemma 2 of reference ^[7].

Lemma 3. Let $p$ be an odd prime, $s$ be any integer with $(s, p) = 1$. Then for any non-principal character $\chi \bmod p$ and any Dirichlet character $\lambda \bmod p$, we have

where $\tau(\chi) = \sum_{a = 1}^{p}\chi(a) e\left(\frac{a}{p}\right)$ denotes the classical Gauss sums.

Proof. See Lemma 1 of reference ^[7].

Lemma 4. Let $p$ be an odd prime, then we have

Proof. See Lemma 5 of reference ^[6].

3.   Proof of Theorems

Now we come to prove our Theorems.

Firstly, we prove Theorem 1. Applying Proposition $1$ and Lemma $1$, we obtain

Then from Lemma $2$ and Lemma $4$, if $\overline{\lambda}\chi = \chi_{0}$, we have

While if $\overline{\lambda}\chi\neq\chi_{0}$, we have

This completes the proof of Theorem 1.

Then we prove Theorem 2. From Proposition $1$ and Lemma $1$, we obtain

Since $p\equiv 1 \bmod 4$, and notice that $\left|\tau\left(\overline{\chi}^{2}\right)\right| = \sqrt{p}$. From Lemma $3$ and Lemma $4$, if $\overline{\lambda\chi}\neq\chi_{0}$, $\overline{\lambda}\chi\neq\chi_{0}$, we have

Similarly, if $\overline{\lambda\chi}\neq\chi_{0}, \ \overline{\lambda}\chi = \chi_{0}$, we have

If $\overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi = \chi_{0}$, we can obtain

So we have

Noting that $x\leq |x|$ holds for any real number $x$, we have

Similarly, if $\overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0}$, we have

This completes the proof of Theorem 2.

Next we turn to prove Theorem 3. Since $p\equiv 3 \bmod 4$, note that $\left(\frac{-1}{p}\right) = \chi_{2}(-1) = -1$, $L(1, \chi_{2}) = \frac{\pi\cdot h_{p}}{\sqrt{p}}$, and $\tau\left(\overline{\chi}^{2}_{2}\right) = -1$. From Lemma $3$ and Lemma $4$, if $\overline{\lambda\chi}\neq\chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0}$, we have

Similarly, if $\overline{\lambda\chi}\neq\chi_{0}, \ \overline{\lambda}\chi = \chi_{0}$, we have

If $\overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi = \chi_{0}$, we can obtain

So we have

Similarly, if $\overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0}$, we have

Combining the fact that

we deduce that if $p\equiv 3 \bmod 8$, then

If $\overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi = \chi_{0}$, we have

Then

Similarly, if $\overline{\lambda\chi} = \chi_{0}, \ \overline{\lambda}\chi\neq\chi_{0}$, we have

This completes the proof of Theorem 3.

Theorem 4 can be derived by the same method. This completes the proof of our Theorems.

4.   Conclusions

In this paper, we obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums and Kloosterman sums (both classical Kloosterman sums and general Kloosterman sums) by applying the properties of Gauss sums and the mean value of Dirichlet $L$-function. But in some cases, unluckily, it is difficult to get the exact formula. So how to get the exact formula in all cases remains open.

Acknowledgments

The authors want to show their great thanks to the anonymous referee for his/her helpful comments and suggestions.

This work is supported by the National Natural Science Foundation of China (No. 11871317, 11926325, 11926321) and the Fundamental Research Funds for the Central Universities (No. GK201802011).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.