Sign of Fourier coefficients of modular forms of half integral weight

TL;DR

This paper proves lower bounds on the number of positive, negative, and sign-changing Fourier coefficients at squarefree integers for half-integral weight modular Hecke eigenforms, advancing understanding of their sign patterns.

Contribution

It establishes new lower bounds for sign changes and counts of Fourier coefficients at squarefree integers in half-integral weight modular forms.

Findings

Lower bounds for positive and negative Fourier coefficients

Lower bounds for sign changes in the sequence

Quantitative results on the distribution of signs

Abstract

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.