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

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 $j=2,3,4.$

Locations

• Proceedings of the American Mathematical Society - View - PDF