The Mean Values of Character Sums and Their Applications

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.

One Kind New Hybrid Power Mean and Its Computational Formulae

“…Note that 2 is a cubic residue modulo p if and only if d is an even number (see Corollary 1 in [18]). So from Teorems 1, 2, and 4, we may immediately deduce the following two corollaries.…”

On the Mean Value of High-Powers of a Special Character Sum Modulo a Prime

“…Many mathematicians have studied the properties of the Legendre symbol and obtained a series of important results (see [1][2][3][4][5][6][7][8][9][10][11][12][13]). Perhaps the most representative properties of the Legendre's symbol are as follows:…”