Asian Journal of Mathematics

Volume 25 (2021)

Number 3

Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions

Pages: 413 – 430

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n3.a5

Authors

Nilanjan Bag (Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India)

Rupam Barman (Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India)

Abstract

The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the 6th and 8th order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.

Keywords

generalized quadratic Gauss sums, $L$-functions, asymptotic formula

2010 Mathematics Subject Classification

11L05, 11M20

Full Text (PDF format)

Received 3 September 2020

Accepted 18 September 2020

Published 14 March 2022