On sums of Hecke eigenvalue squares over primesin very short intervals
Volume 205 / 2022
Acta Arithmetica 205 (2022), 309-321
MSC: Primary 11F30; Secondary 11N05, 11F72.
DOI: 10.4064/aa210913-31-8
Published online: 27 October 2022
Abstract
Let $\eta \gt 0$ be a fixed positive number, and let $N$ be a sufficiently large number. We study the second moment of the sum of Hecke eigenvalues over primes in short intervals (of length $\eta \log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke–Maass cusp forms for ${\rm SL}(2,\mathbb {Z})$ and ${\rm SL}(3,\mathbb {Z}).$ By applying the Hardy–Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.
