Oscillatory behavior and equidistribution of signs of Fourier coefficients of cusp forms

TL;DR

This paper investigates the oscillatory patterns and distribution of signs in Fourier coefficients of cusp forms, exploring their behavior across different subfamilies and weights using advanced number theory techniques.

Contribution

It introduces new insights into the sign distribution and oscillatory behavior of Fourier coefficients for both integral and half-integral weight cusp forms.

Findings

Signs of Fourier coefficients are equidistributed in certain subfamilies.

Oscillatory behavior varies with the weight and type of cusp form.

Results extend understanding of Fourier coefficient distributions in automorphic forms.

Abstract

In this paper, we discuss questions related to the oscillatory behavior and the equidistribution of signs for certain subfamilies of Fourier coefficients of integral weight newforms with a non-trivial nebentypus as well as Fourier coefficients of eigenforms of half-integral weight reachable by the Shimura correspondence.