The first moment of Salie sums

TL;DR

This paper investigates the asymptotic behavior of Salie sums over arithmetic progressions, revealing a positivity bias, using Kuznetsov formula and explicit modular forms basis.

Contribution

It provides the first asymptotic analysis of Salie sums and develops an explicit orthogonal basis for half-integral weight modular forms.

Findings

Salie sums tend to be positive, indicating a bias.

Derived an explicit asymptotic formula for Salie sums.

Constructed an explicit orthogonal basis for half-integral weight modular forms.

Abstract

The main objective of this article is to study the asymptotic behavior of Salie sums over arithmetic progressions. We deduce from our asymptotic formula that Salie sums possess a bias of being positive. The method we use is based on Kuznetsov formula for modular forms of half integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half integral weight.