On Hecke Eigenvalues at Piatetski-Shapiro Primes

Abstract

Let $\hat{I}»(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C>0$ depending on the cusp form and every fixed $c$ in the range $1<c<8/7$, the mean value of $\hat{I}»(p)$ is $\ll \exp (-C\sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n\le N$.

24 pages, with updated reference and minor revisions