Selberg sums - a new perspective

TL;DR

This paper explores Selberg sums over finite fields, their properties, and their connections to Gauss sums and polynomial discriminants, offering a new perspective on their mathematical structure and applications.

Contribution

The paper introduces a new perspective on Selberg sums, analyzing their properties and their relation to Gauss sums and polynomial discriminants in finite fields.

Findings

Selberg sums relate to the distribution of Gauss sums.

Connections between Selberg sums and polynomial discriminants are established.

Basic properties of Selberg sums are formulated and analyzed.

Abstract

Selberg sums are the analogues over finite fields of certain integrals studied by Selberg in in 1940s. The original versions of these sums were introduced by R.J.Evans in 1981 and, following an elegant idea of G.W.Anderson in 1991 they were evaluated by Anderson, Evans and P.B.~van~Wamelen. In 2007 the author noted that these sums and certain generalizations of them appear in the study of the distribution of Gauss sums over a rational function field over a finite field. The distribution of Gauss sums is closely related to the distribution of the values of the discriminant of polynomials of a fixed degree. Here we shall take this up further. The main goal here is to establish the basic properties of Selberg sums and to formulate the problems which arise from this point of view.