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 $\re s>\tfrac{1}{2}$. This exhibits a high fluctuation of the coefficients at squarefree integers.
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Y.-J Jiang, Y.-K Lau, G.-S Lü, Emmanuel Royer, J. Wu. On Fourier coefficients of modular forms of half integral weight at squarefree integers. Mathematische Zeitschrift, Springer, 2019, 293 (1-2), pp.789-808. ⟨10.1007/s00209-018-2191-1⟩. ⟨hal-01280172v3⟩

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