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On the asymptotics of the shifted sums of Hecke eigenvalue squares.
- Source :
-
Forum Mathematicum . Mar2023, Vol. 35 Issue 2, p297-328. 32p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *EIGENVALUES
*HARDY-Littlewood method
*CUSP forms (Mathematics)
*SQUARE
*INTEGERS
Subjects
Details
- Language :
- English
- ISSN :
- 09337741
- Volume :
- 35
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Forum Mathematicum
- Publication Type :
- Academic Journal
- Accession number :
- 162207188
- Full Text :
- https://doi.org/10.1515/forum-2020-0359