Sign Changes of Coefficients and Sums of Coefficients of L-Functions

TL;DR

This paper generalizes methods for detecting sign changes in sequences of complex numbers, specifically applied to coefficients of various L-functions, demonstrating their sign-changing behavior in multiple contexts.

Contribution

It extends the axiomatization for sign changes to complex sequences and applies it to coefficients of L-functions from different automorphic forms.

Findings

Sign changes occur in sequences of coefficients of GL(2) and GL(3) L-functions.

Sign changes are also shown in partial sums of these coefficients.

The results unify and extend previous sign change results for automorphic forms.

Abstract

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL(2) holomorphic cusp forms, GL(2) Maass forms, and GL(3) Maass forms. Building on previous works by the authors, we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL(2) holomorphic cusp forms and Maass forms.