On the asymptotics of the shifted sums of Hecke eigenvalue squares

Author:

Kim Jiseong1ORCID

Affiliation:

1. Department of Mathematics , University at Buffalo , 244 Mathematics Building Buffalo , NY 14260-2900 , USA

Abstract

Abstract The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for X 2 3 + ϵ < H < X 1 - ϵ {X^{\frac{2}{3}+\epsilon}<H<X^{1-\epsilon}} there are constants B h {B_{h}} such that X n 2 X λ f ( n ) 2 λ f ( n + h ) 2 - B h X = O f , A , ϵ ( X ( log X ) - A ) \sum_{X\leq n\leq 2X}\lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,% \epsilon}(X(\log X)^{-A}) for all but O f , A , ϵ ( H ( log X ) - 3 A ) {O_{f,A,\epsilon}(H(\log X)^{-3A})} integers h [ 1 , H ] {h\in[1,H]} where { λ f ( n ) } n 1 {\{\lambda_{f}(n)\}_{n\geq 1}} are normalized Hecke eigenvalues of a fixed holomorphic cusp form f. Our method is based on the Hardy–Littlewood circle method. We divide the minor arcs into two parts m 1 {m_{1}} and m 2 {m_{2}} . In order to treat m 2 {m_{2}} , we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat m 1 {m_{1}} , we apply Parseval’s identity and Gallagher’s lemma.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Reference23 articles.

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