Moments and equidistributions of multiplicative analogues of Kloosterman sums

TL;DR

This paper studies multiplicative analogues of Kloosterman sums, deriving asymptotic formulas for their moments and establishing an arcsine law, with implications for moments weighted by special L-values.

Contribution

It introduces new asymptotic formulas for moments of multiplicative Kloosterman analogues and proves an arcsine law using the method of moments.

Findings

Asymptotic formulas for moments of the character sums.

Establishment of an arcsine distribution law.

Asymptotic results for moments weighted by special L-values.

Abstract

We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the above character sum as the character runs over all non-trivial multiplicative characters mod $p.$ Moreover, an arcsine law is also established as a consequence of the method of moments. The evaluations of these moments also allow us to obtain asymptotic formulae for moments of such character sums weighted by special $L$-values (at $1/2$ and $1$).