The average behavior of Fourier coefficients of cusp forms over sparse sequences

Lao, Huixue; Sankaranarayanan, Ayyadurai

doi:10.1090/S0002-9939-09-09855-4

The average behavior of Fourier coefficients of cusp forms over sparse sequences
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by Huixue Lao and Ayyadurai Sankaranarayanan PDF

Proc. Amer. Math. Soc. 137 (2009), 2557-2565 Request permission

Abstract:

Let $\lambda (n)$ be the $n$th normalized Fourier coefficient of a holomorphic Hecke eigenform $f(z)\in S_{k}(\Gamma )$. In this paper we are interested in the average behavior of $\lambda ^2(n)$ over sparse sequences. By using the properties of symmetric power $L$-functions and their Rankin-Selberg $L$-functions, we are able to establish that for any $\varepsilon >0$, \[ \sum _{n \leq x}\lambda ^2(n^j)=c_{j-1} x+O\left (x^{1-\frac {2}{(j+1)^2+2}+\varepsilon }\right ),\] where $j=2,3,4.$

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