A short note on sign changes

TL;DR

This paper provides quantitative results on the number of sign changes in Fourier coefficients of modular forms and generalized modular functions, advancing understanding of their oscillatory behavior.

Contribution

It offers new quantitative bounds on sign changes for Fourier coefficients of Hecke eigenforms and generalized modular functions, extending previous qualitative results.

Findings

Quantitative bounds on sign changes for Fourier coefficients of cusp forms

Results on sign changes of $q$-exponents for generalized modular functions

Extension of sign change analysis to functions on congruence subgroups

Abstract

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group $SL_2(\mathbb{Z})$. We also prove a similar kind of quantitative result for the number of sign changes of the $q$-exponents $c(p) (p {vary over primes})$ of certain generalized modular functions for the congruence subgroup $\Gamma_0(N)$, where $N$ is square-free.