On Products of Fourier Coefficients of Cusp Forms

TL;DR

This paper investigates the behavior of products of Fourier coefficients of elliptic cusp forms, focusing on their non-vanishing and non-negativity properties for fixed shifts.

Contribution

It provides new insights into the non-vanishing and positivity of products of Fourier coefficients of cusp forms for fixed shifts.

Findings

Results on non-vanishing of coefficient products

Conditions for non-negativity of coefficient products

Theoretical bounds and implications

Abstract

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, $a(n)a(n + r)$ $(n \geq 1)$ for a fixed positive integer $r$, concerning both non-vanishing and non-negativity.