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

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.