Average behavior of Fourier coefficients of cusp forms

Lü, Guangshi

doi:10.1090/S0002-9939-08-09741-4

Average behavior of Fourier coefficients of cusp forms
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Proc. Amer. Math. Soc. 137 (2009), 1961-1969 Request permission

Abstract:

Let $a_0(n)$ and $b_0(n)$ be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms $f(z)\in S_{2k}(\Gamma )$ and $\varphi (z)\in S_{2l}(\Gamma )$ respectively. In 1999, Fomenko studied the following average sums of $a_0(n)$ and $b_0(n)$: \[ \sum _{n \leq x}a_0(n)^3, \quad \sum _{n \leq x}a_0(n)^2b_0(n), \quad \sum _{n \leq x}a_0(n)^2b_0(n)^2, \quad \sum _{n \leq x}a_0(n)^4. \] In this paper, we are able to improve on Fomenko’s results.

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