On sum of Hecke eigenvalue squares over primes in very short intervals

TL;DR

This paper investigates the average second moment of Hecke eigenvalues over primes in very short intervals for various automorphic forms, providing explicit mean value calculations under the Hardy-Littlewood prime 2-tuples conjecture.

Contribution

It extends the analysis of Hecke eigenvalue sums to very short intervals and generalizes results to multiple types of automorphic forms, with explicit mean value formulas.

Findings

Derived the second moment of Hecke eigenvalues over primes in short intervals.

Generalized results to Hecke-Maass cusp forms for different groups.

Calculated exact mean values assuming the Hardy-Littlewood prime 2-tuples conjecture.

Abstract

Let $\eta>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $\eta \log N$) 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 $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.