The Mean Values of Character Sums and Their Applications

Jiafan Zhang,Yuanyuan Meng

doi:10.3390/math9040318

Abstract

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

Highlights

IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
2 is a cubic residue modulo p if and only if there are two integers d1 and b1 such that p = d21 + 27 · b12, where d1 is uniquely determined by d1 ≡ −1(mod3)
Our results reveal the value distribution properties of the character sums of polynomials

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations This function occupies a very important position in elementary number theory and analytic number theory. B=0 a=0 where f (x, y) is an integer coefficient polynomial of x and y, p is an odd prime, χ is any non-principal character modulo p. 2 is a cubic residue modulo p if and only if there are two integers d1 and b1 such that p = d21 + 27 · b12, where d1 is uniquely determined by d1 ≡ −1(mod3) Our results reveal the value distribution properties of the character sums of polynomials. For any prime p with p ≡ 1(mod 6) and any non-principal character χ modulo p, whether there is an exact calculation formula for the 2k-th power mean p−1 p−1. This describes the distribution properties of the quadratic residue in the form of a3 + 2 modulo p from a different perspective

Several Lemmas

Proofs of the Theorems

Findings

Conclusions