An explicit evaluation of $\nth{10}$-power moment of quadratic Gauss sums and some applications

TL;DR

This paper provides an explicit estimate of a complex multi-variable character sum related to quadratic Gauss sums and applies this to analyze the tenth power mean value of these sums, advancing understanding in number theory.

Contribution

The paper introduces a new explicit estimate for a multi-variable character sum involving quadratic residues and applies it to evaluate the tenth power mean of generalized quadratic Gauss sums.

Findings

Derived an explicit estimate for a multi-variable character sum.

Analyzed the tenth power mean value of quadratic Gauss sums.

Enhanced methods for studying character sums and their moments.

Abstract

In this paper we have estimated one multi-variable character sum \begin{align*} \sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{c=2}^{p-2}\sum_{d=1}^{p-1}\left(\frac{a^2-b^2}{p}\right)\left(\frac{b^2-1}{p}\right) \left(\frac{c^2-d^2}{p}\right)\left(\frac{d^2-1}{p}\right)\left(\frac{a^2c^2-1}{p}\right), \end{align*} for odd prime $p$. With the help of our estimate of the above character sum, we have studied the tenth power mean value of generalized quadratic Gauss sums using estimates for character sums and analytic methods.