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

Y.-J Jiang (1) , Y.-K Lau (2) , Emmanuel Royer (3) , J Wu (4)

1 TELECOM - ENAC - Equipe télécommunications
2 Department of Mathematics
3 LMBP - Laboratoire de Mathématiques Blaise Pascal
4 IECL - Institut Élie Cartan de Lorraine

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 $\re s>\tfrac{1}{2}$. This exhibits a high fluctuation of the coefficients at squarefree integers.

Domaines: Théorie des nombres [math.NT]

Y.-J Jiang, Y.-K Lau, Emmanuel Royer, J Wu. On Fourier coefficients of modular forms of half integral weight at squarefree integers. 2016. ⟨hal-01280172v2⟩