Mean values of character sums analogue of Kloosterman sums

TL;DR

This paper investigates the average behavior of a character sum analogue of Kloosterman sums, focusing on mean square values and bilinear forms, extending understanding of these sums in number theory.

Contribution

It introduces new results on mean square values and bilinear forms for character sum analogues of Kloosterman sums, advancing the theoretical framework in this area.

Findings

Established mean square value estimates for the sums.

Derived bounds for bilinear forms involving these sums.

Extended classical results to character sum analogues.

Abstract

Let $q$ be a positive integer, $\chi$ a nontrivial character mod $q$, $\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shall study the character sum analogue of the well-known Kloosterman sum,\[\sum_{\substack{a\in\mathcal{I} \gcd(a,q)=1}}\chi(ma+n\overline{a}),\] where $\overline{a}$ is the multiplicative inverse of $a\bmod q$. The mean square values and bilinear forms for such sums are proved.