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

2021/08/10 by Jiseong Kim, Kim, Jiseong
Mathematics · #11A41 #11F30 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2108.04968

openalex publication_date 2021/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let η>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 ηlog 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,ℤ) and SL(3,ℤ). By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.

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