On Fourier coefficients of modular forms of half integral weight at squarefree integers

TL;DR

This paper proves that the Dirichlet series of Fourier coefficients of half-integral weight Hecke eigenforms at squarefree integers extends holomorphically to a half-plane, revealing significant fluctuation in these coefficients.

Contribution

It establishes the analytic continuation of the Dirichlet series at squarefree integers for half-integral weight Hecke eigenforms, highlighting their coefficient fluctuations.

Findings

Dirichlet series extends holomorphically to Re s > 1/2

Fourier coefficients exhibit high fluctuation at squarefree integers

Analytic properties relate to the behavior of half-integral weight forms

Abstract

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s\textgreater{}\tfrac{1}{2}$. This exhibits a high fluctuation of the coefficients at squarefree integers.