The fourth power mean of the generalized quadratic Gauss sums associated with some Dirichlet characters

1.   Introduction

Let $q\in \mathbf{Z}^{+}$ with $q\geq 2$. $\lambda$ denotes a Dirichlet character modulo $q$. For any $m, n\in\mathbf{Z}$, the generalized quadratic Gauss sums $\mathcal{G}(m, n, \lambda; q)$ is defined as follows:

where $\text{e}_{q}(x) = \text{exp}\left(2\pi \text{i}x/ q\right)$ and $\text{i}$ is the imaginary unit (i.e., $\text{i}^2 = -1$).

If $m = 0$ and $n = 1$, then we call $\mathcal{G}(0, 1, \lambda; q)$ a classical Gauss sum and denote it as $\mathcal{G}(\lambda)$. Gauss sums have had an important effect on both cryptography and the analytic number theory. The analytic number theory and cryptography will greatly benefit from any significant advancements made in this area, so the study of the properties of $\mathcal{G}(m, n, \lambda; q)$ is a meaningful work. Up to now, many researchers have studied the calculation and estimation of the high-th power mean of $\mathcal{G}(m, n, \lambda; q)$, which can be roughly divided into three types. They are

For example, W. P. Zhang ^[13] proved the identities

where $p$ is an odd prime and $m$ is any integer with $(m, p) = 1$, $\lambda_{2}$ denotes the Legendre's symbol modulo $p$, and $\phi(x)$ is the Euler function.

In 2020, N. Bag and R. Barman ^[2] proved that for odd prime $p$ and any $m\in\mathbf{Z}$ with $(m, p) = 1$, there are the asymptotic formulae

and

Over the next two years, N. Bag, A. Rojas-Le$\acute{ \text{o}}$n, and W. P. Zhang ^[3,4] proved not only the asymptotic formula for the $10$-th power mean of $\mathcal{G}(m, 0, \lambda; p)$, but also the asymptotic formula for the any $2k$-th power mean of $\mathcal{G}(m, 0, \lambda; p)$. The results are as follows:

X. X. Li and Z. F. Xu ^[9] studied the fourth power mean of the generalized quadratic Gauss sums, and proved the following result: For an odd prime $p$ and a character $\lambda$ modulo $p$, there are the identities

where $\bar{t}$ is the inverse of $t$ modulo $p$ (i.e., $\bar{t}\cdot t\equiv1\bmod p$), and $\lambda_{0}$ denotes principal characters mod $p$.

X. Y. Liu and W. P. Zhang ^[10] proved that for any odd prime $p$ with $3\nmid \phi(p)$, one has the identity

X. X. Lv and W. P. Zhang ^[11] obtained the identities, but there may be a bit of a miscalculation in this result, and the correct result should be

Some related work can also be found in ^[6,12,14,16]. We will not list them all here. It is worth noting that for the power mean of $\mathcal{G}(m, 1, \lambda; p)$,

no one seems to have studied it so far; at least we have not seen any valid conclusions. Through research, we have found that even taking $k = 2$ in (1.1), it is difficult to get the exact results for general Dirichlet characters. Therefore, we settle for the second: select some special Dirichlet characters to get the exact result.

Throughout the article, we fix a few notations. We use $f = O(g)$ to denote $|f|\leq c g$ for some positive constant $c$. We use a $\lambda_{k}$ to denote $k$-order Dirichlet character modulo $p$ (i.e., $\lambda_{k}^{k} = \lambda_{0}$), where $\lambda_{0}$ denotes principal characters.

Theorem 1. Let $p$ be an odd prime with $3\mid \phi(p)$. Then, for any $3$-order character $\lambda_{3}$ modulo $p$ with $\lambda_{3}(2) = 1$, we have the identities

where $d$ is uniquely determined by $4p = d^2+27b^2$ and $d\equiv 1\bmod 3$.

Theorem 2. Let $p$ be an odd prime with $4\mid \phi(p)$. Then, for any $4$-order character $\lambda_{4}$ modulo $p$, we have the identity

where $\alpha(p)$ is defined by $\alpha(p) = \sum_{s = 1}^{\frac{p-1}{2}}\lambda_{2}(s+\overline{s})$.

From these two theorems, we may immediately get the following corollary:

Corollary 1. Let $p$ be an odd prime. Then, for any Dirichlet character $\lambda$ with $\lambda\neq\lambda_{0}$, we have the asymptotic formula

2.   Several lemmas

The primary goal of this section is to introduce several necessary lemmas and their proofs for the paper's main results. In the proof of these lemmas, we use the properties of Gauss sums and the reduced (complete) residue system, as well as the properties and definitions of $k$-order character, odd character, and even character. References ^[1,8,15] have further details; we will not explain them here.

Lemma 1. Let $p$ be an odd prime. Then, for any Dirichlet character $\lambda$ modulo $p$ with $\lambda\neq\lambda_{0}$, we have the identity

Proof. We used Dirichlet character sum $\sum\limits_{s = 0}^{p-1}\lambda\left(s^2-1\right)$ as a bridge to analyze it from different perspectives. On one hand, there is

On the other hand, for any integer $t$ with $(t, p) = 1$, the identity

there is

Note that $\overline{\mathcal{G}(\lambda)} = \lambda(-1)\cdot \mathcal{G}\left(\overline{\lambda}\right)$, $\mathcal{G}^2\left(\lambda_2\right) = \lambda_2(-1)\cdot p$, $\mathcal{G}(\lambda)\cdot \overline{\mathcal{G}(\lambda)} = p$, from (2.1) and (2.2) we have the identities

This proves Lemma 1. □

Lemma 2. Let $p$ be an odd prime, and let $\lambda$ be any Dirichlet character modulo $p$ with $\lambda\neq\lambda_{0}$. If $\lambda$ is an odd character modulo $p$, that is, $\lambda(-1) = -1$, then we have

If $\lambda$ is an even character modulo $p$, that is, $\lambda(-1) = 1$, then we have

where $\psi$ is a Dirichlet character modulo $p$ such that $\lambda = \psi^2$.

Proof. If $\lambda$ is an odd character modulo $p$, this is $\lambda(-1) = -1$, then we have

Hence

If $\lambda$ is an even character modulo $p$, then there exists a Dirichlet character $\psi$ such that $\lambda = \psi^2$, and we obtain

In Lemma 1, we have

Similarly,

Now combine (2.3)–(2.5) to obtain identity

This proves Lemma 2. □

Lemma 3. Let $p$ be an odd prime. Then, for any Dirichlet character $\lambda$ modulo $p$ with $\lambda\neq\lambda_{0}$, we have the identity

Proof. Note that the conditions $s^2+t^2\equiv u^2+1\bmod p$ and $s+t\equiv u+1\bmod p$ equivalent to $s+t\equiv u+1\bmod p$ and $st\equiv u\bmod p$, we obtain

This proves Lemma 3. □

Lemma 4. Let $p$ be an odd prime with $3\mid\phi(p)$. Then, we have the identity

Proof. See This is consequence of ^[5](pp. 114). □

Lemma 5. Let $p$ be an odd prime with $4\mid\phi(p)$. Then, we have the identity

Proof. See Lemma 2.2, Section 2 in ^[7](pp. 1253).□

3.   Proofs of the theorems

Now we apply Lemmas 1–3 to complete the proof of our Theorem 1 and Theorem 2. In fact, note that the trigonometrical identities

From the properties of the reduced residue system modulo $p$, we have

If $\lambda(-1) = -1$, then from (3.1), Lemmas 2 and 3, we have

Similarly, if $\lambda(-1) = 1$, then there exists a Dirichlet character $\psi$ such that $\lambda = \psi^2$, and we obtain

If $3\mid \phi(p)$ and $\lambda_{3}(2) = 1$, then $\lambda_{3}(4) = 1$ and $\lambda_{3} = \overline{\lambda_{3}}^2$. Taking $\lambda = \lambda_{3}$ and $\psi = \overline{\lambda_{3}}$, from (3.2) and identity

we have

This proves Theorem 1.

If $4\mid \phi(p)$, since $\lambda_2(4) = 1$, $\lambda_2(-1) = 1$, $\mathcal{G}(\lambda_2) = \sqrt{p}$ and $\lambda_2 = \lambda_{4}^2$. Taking $\lambda = \lambda_2$ in (3.2) and $\psi = \lambda_{4}$, from (3.2) and identity

we have

which implies Theorem 2 and Corollary 1.

4.   Conclusions

Firstly, for any integer $k\geq 3$ in (1.1), it is difficult to obtain a corresponding result by using our methods, which is an open problem. Finally, our Theorem 1 only obtained a simple identity for the case where $\lambda_{3}(2) = 1$. What would happen if $\lambda_{3}(2)\neq1$.

Author contributions

All authors have equally contributed to this work, and all authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Hainan Provincial Natural Science Foundation of China (123RC473) and the Natural Science Foundation of China (12126357).

The authors would like to thank the referees for their very helpful and detailed comments.

Conflict of interest

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