Some estimate of character sums and its applications

Jianghua Li,Di Han

doi:10.1186/1029-242x-2013-328

Abstract

The main purpose of this paper is using the elementary methods and the properties of Gauss sums to give a sharp estimate for some character sums. Then using this estimate to prove the existence of some special primitive roots modp, an odd prime, and to prove that for any integer n with (n,p)=1, if p is large enough, then there exist two primitive roots α and β of p such that both α+nβ and n α ¯ + β ¯ are also primitive roots of p, where a ¯ satisfies a ¯ a≡1modp. Let N(n,p) denote the number of all pairs (α,β) of primitive roots of p such that both α+nβ and n α ¯ + β ¯ are also primitive roots of p. Then we can give an interesting asymptotic formula for N(n,p).MSC:11M20.

Highlights

If q has a primitive root, each reduced residue system mod q can be expressed as a geometric progression
This gives a powerful tool that can be used in problems involving reduced residue systems
Authors’ contributions JL carried out the sharp estimate for some character sums and give an asymptotic formula for N(n, p)

Summary

Introduction

Primitive roots exist only for the following several cases: q = , , , pα, pα, where p is an odd prime and α ≥ . Let N(n, p) denote the number of all pairs (α, β) of primitive roots of p such that both α + nβ and nα + βare primitive roots of p. We shall use the elementary methods and estimate for character sums to study this problem, and prove the following conclusion.

Objectives

Results

Conclusion