On sums of Hecke-Maass eigenvalues squared over primes in short intervals

TL;DR

This paper establishes a uniform estimate for sums of squared Hecke--Maass eigenvalues over primes in short intervals, extending classical prime number theorems to automorphic forms with effective constants.

Contribution

It provides a new uniform estimate for automorphic eigenvalues over primes in short intervals, inspired by classical prime number theorems and recent advances in automorphic representation theory.

Findings

Uniform estimate for sums of eigenvalues over primes in short intervals

Constants are universal and effectively computable

Improved bounds and detailed explanations in the latest version

Abstract

We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those implicit, are all universal and effectively computable, although we pay no particular attention to numerical precision. In this third version, we have made an improvement upon the main assertion and added greater details. In this new version some augmentation is made concerning the bounds for logarithmic derivatives of relevant symmetric L-functions. It is, however, only to make our argument more accessible.