On Hecke Eigenvalues at Piatetski-Shapiro Primes

TL;DR

This paper investigates the average behavior of Fourier coefficients of cusp forms at Piatetski-Shapiro primes, showing exponential decay in the mean value for certain exponents c between 1 and 8/7.

Contribution

It establishes a decay estimate for Fourier coefficients at Piatetski-Shapiro primes within a specific range of c, extending understanding of their distribution at special prime sequences.

Findings

Mean value of Fourier coefficients decays exponentially for primes of form [n^c] with 1 < c < 8/7.

The decay rate depends on a constant C related to the cusp form.

Results hold uniformly for all n up to N.

Abstract

Let $\lambda(n)$ 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 $\lambda(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \leq N$.