On the fourth power mean of the general k th Kloosterman sums

Xiaoyan Guo,Guohua Geng,Xiaowei Pan

doi:10.1186/1687-1847-2014-103

Abstract

Let q > 2 be an integer, and let χ be a Dirichlet character modulo q. For integers m, n and k, the general kth Kloosterman sum S(m, n, k, χ ; q )i s def ined by

Highlights

Let q > be an integer, and let χ be a Dirichlet character modulo q
), where denotes the summation over all a with (a, q) = 1, e(y) = e2πiy, and a is the inverse of a modulo q such that aa ≡ 1 mod q and 1 ≤ a ≤ q
In this paper we further study the fourth power mean χ mod q q m=1

Summary

Introduction

Let q > be an integer, and let χ be a Dirichlet character modulo q. For arbitrary integers m and n, the general Kloosterman sum S(m, n, χ ; q) is defined by q ma + na. N and k, the general kth Kloosterman sum S(m, n, k, χ ; q) is defined by q mak + nak. Let α and k be positive integers with d = (k, p – ) and (k, pα– ) = pδ. Let p be an odd prime, n be any integer. Let α and k be positive integers. Let q > be an odd number, n be an integer with (n, q) = , and let k be a positive integer.

Then we have q

Therefore pα pα