On the asymptotics of the shifted sums of Hecke eigenvalue squares

Jiseong Kim

doi:10.1515/forum-2020-0359

Article

On the asymptotics of the shifted sums of Hecke eigenvalue squares

Jiseong Kim

Published/Copyright: January 30, 2023

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From the journal Forum Mathematicum Volume 35 Issue 2

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 there are constants 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 )

for all but O f , A , ϵ ⁢ ( H ⁢ ( log ⁡ X ) - 3 ⁢ A ) integers h ∈ [ 1 , H ] where { λ f ⁢ ( n ) } n ≥ 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 and m 2 . In order to treat 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 , we apply Parseval’s identity and Gallagher’s lemma.

Keywords: Hecke eigenvalue; shifted convolution

MSC 2010: 11F30

Communicated by Valentin Blomer

Acknowledgements

The author would like to thank his advisor Xiaoqing Li, for suggesting this problem and giving helpful advice. The author is grateful to the referees for a careful reading of the paper and lots of invaluable suggestions.

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Received: 2020-12-24

Revised: 2022-12-11

Published Online: 2023-01-30

Published in Print: 2023-03-01

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https://doi.org/10.1515/forum-2020-0359

Keywords for this article

Hecke eigenvalue; shifted convolution