A certain new Gauss sum and its fourth power mean

1.   Introduction

Let $q\geq 3$ is a positive integer. For any integral coefficient polynomial $f(x)$ and any Dirichlet character $\chi\bmod q$, we define a generalized Gauss sums $G(f, \chi; q)$ as

where as usual, $e(y) = e^{2\pi iy}$.

If $f(x) = x$, then $G(f, \chi; q) = \tau(\chi)$ is the classical Gauss sums. That is,

A very important property of $\tau(\chi)$ is that if $\chi$ is a primitive character $\bmod\ q$, then we have the identity $|\tau(\chi)| = \sqrt{q}$ (see Theorem 8.15 in ^[1]).

If $f(x) = ax^2+bx$, then $G(f, \chi; q)$ becomes the generalized quadratic Gauss sums. Many authors have studied its properties and obtained interesting results. For example, if $q = p$ is an odd prime, then from A. Weil's important work ^[2] we can get the estimate

for all integers $a$ and $b$ with $(a, b, p) = 1$.

Let $p$ be an odd prime, $\alpha$ be a positive integer $\alpha \geq 2$, and $\lambda$ be a primitive character modulo $\ p^{\alpha}$. Zhang Wenpeng and Lin Xin ^[3] proved that for any integer $n$ with $(n, p) = 1$, we have

If $\lambda$ is any non-primitive character modulo $p^{\alpha}$, then we have the identity

where $\phi(n)$ denotes the Euler function.

Li Xiaoxue and Xu Zhefeng ^[4] also studied the special case $q = p$, an odd prime, and obtained several interesting identities.

On the other hand, Lv Xingxing and Zhang Wenpeng ^[5] introduced a new sum analogous to Kloosterman sum as follows:

where $m, \ n$ and $r$ are integers, $\overline{a}$ denotes $a\cdot \overline{a}\equiv 1\bmod q$.

Using properties of the character modulo $p$ and analytic methods, they proved the following results: For any odd prime $p$ with $p\equiv 3\bmod 4$ and integers $m$ and $n$ with $(mn, p) = 1$, one has the identity

and

From the second formula and the estimate for Kloosterman sums, Lv Xingxing and Zhang Wenpeng ^[5] deduced the asymptotic formula

Shane Chern ^[6] also studied the properties of $K(m, n, r, \chi; q)$, and obtained the identity

where $p$ is an odd prime, $n$ and $k$ are integers with $(nk, p) = 1$.

Some other papers related to Gauss sums and two-term exponential sums can also be found in ^{[5,6,7,8,9,10,11,12,13]}, but we will not go into further detail here.

Inspired by ^[5] and ^[6], it is natural to study the $2k$-th power means

To the best knowledge of the authors, sums of the type (1.2) have not been previously investigated but the authors think that this problem is meaningful, and can be viewed as a new combination of additive and multiplicative functions.

In this paper, we will use the number of the solutions of some congruence equations $\bmod\ p$ and the properties of the Legendre symbol to study the problem of calculating (1.2) with $k = 2$, and give two exact calculating formulae according to $p\equiv 1\bmod 4$ or $p\equiv 3\bmod 4$. That is, we will prove the following results:

Theorem 1. Let $p$ be an odd prime with $p\equiv 3\bmod 4$, then we have

Theorem 2. Let $p$ be an odd prime with $p\equiv 1\bmod 4$, then we have

Some notes: For $k\geq 3$, whether there exists an exact calculating formula for (1.2) is an open problem. It is also interesting to ask whether a general formula analogous to Theorems 1 and 2 can be obtained for all integers $q\geq 3$ and $k = 2$.

2.   Some basic lemmas

In this section, we first give four basic lemmas, which are necessary in the proof of our theorems. Of course, in order to prove these lemmas, we need some knowledge of elementary and analytic number theory. They can be found in reference ^[1], and we do not repeat them all here. First we have the following:

Lemma 1. For any odd prime $p$, we have the identity

Proof. Note that if $a, \ b, \ c$ pass through a reduced residue system $\bmod\ p$, then $da$, $db$ and $cd$ also pass through a reduced residue system $\bmod\ p$, providing $(d, p) = 1$. So from these properties we have

This proves Lemma 1.

Lemma 2. If $p$ is an odd prime, then we have the identity

Proof. It is clear that if $d$ passes through a reduced residue system $\bmod\ p$, then $\overline{d}$ also passes through a reduced residue system $\bmod\ p$. From the properties of the reduced residue system and quadratic residue $\bmod\ p$ we have

It is clear that if $a$ passes through a complete residue system $\bmod\ p$, then $a+1$ also passes through a complete residue system $\bmod\ p$. So we make the change of variables $a$ to $a-1$ and $b$ to $b-1$ modulo $p$ to change the congruence from $ab+a+b\equiv c \bmod p$ to $ab\equiv c+1\bmod p$. Then from (2.1) and the orthogonality property for the Dirichlet character modulo $p$ we have

This proves Lemma 2.

Lemma 3. If $p$ is an odd prime, then we have the identity

Proof. It is clear that the conditions $ab\equiv cd\bmod p$ and $ab+cd\equiv 2\bmod p$ are equivalent to $ab\equiv cd\equiv 1\bmod p$. So we have

This proves Lemma 3.

Lemma 4. If $p$ is an odd prime with $p\equiv 1\bmod 4$, then we have the identity

Proof. Since $p\equiv 1\bmod 4$, there exists an integer $k$ satisfying the congruence $(\pm k)^2\equiv - 1\bmod p$. In fact, these are all such solutions modulo $p$. Choosing $0\leq h\leq p-1$ for which $a+b\equiv c+d\equiv h\bmod p$, we have

3.   Proof of the theorems

In this section, we use the four basic lemmas of the previous section to prove our theorems. First we prove Theorem 1. If $p\equiv 3\bmod 4$, then $\left(\frac{-1}{p}\right) = -1$ and $p\nmid \left(a^2+1\right)$ for all $0\leq a\leq p-1$. Then from Lemma 1, Lemma 2, Lemma 3, the trigonometric identity

and the orthogonality of characters $\bmod\; p$

we have

This proves Theorem 1.

Now we prove Theorem 2. Note that if $p\equiv 1\bmod 4$, then $\left(\frac{-1}{p}\right) = 1$. If $a^2+1\equiv 0\bmod p$, then for any character $\chi\bmod p$, we have $\chi\left(a^2+1\right) = 0$. So from Lemma 4 and the methods of proving Theorem 1 we have

This completes the proof of all of our results.

Acknowledgments

The authors express their sincere thanks to the reviewer for pointing out the errors in our paper.

This work is supported by the Xian Science and Technology Planning Project (2019KJWL28) and N. S. F. (11771351) of P. R. China.

Conflict of interest

The authors declare no conflict of interest.