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On the asymptotics of the shifted sums of Hecke eigenvalue squares

  • Jiseong Kim ORCID logo EMAIL logo
Veröffentlicht/Copyright: 30. Januar 2023
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 35 Heft 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.

References

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Received: 2020-12-24
Revised: 2022-12-11
Published Online: 2023-01-30
Published in Print: 2023-03-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the asymptotics of the shifted sums of Hecke eigenvalue squares
  3. On the splitting conjecture in the hybrid model for the Riemann zeta function
  4. Almost Dedekind domains without radical factorization
  5. Groups in which the centralizer of any non-central primary element is maximal
  6. GKM actions on cohomogeneity one manifolds
  7. Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
  8. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  9. Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
  10. Measures of noncompactness of interpolated polynomials
  11. The homotopy category of acyclic complexes of pure-projective modules
  12. The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
  13. Sharp Li–Yau inequalities for Dunkl harmonic oscillators
  14. A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
  15. Free polynilpotent groups and the Magnus property
https://doi.org/10.1515/forum-2020-0359
Schlagwörter für diesen Artikel
Hecke eigenvalue; shifted convolution

Artikel in diesem Heft

  1. Frontmatter
  2. On the asymptotics of the shifted sums of Hecke eigenvalue squares
  3. On the splitting conjecture in the hybrid model for the Riemann zeta function
  4. Almost Dedekind domains without radical factorization
  5. Groups in which the centralizer of any non-central primary element is maximal
  6. GKM actions on cohomogeneity one manifolds
  7. Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
  8. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  9. Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
  10. Measures of noncompactness of interpolated polynomials
  11. The homotopy category of acyclic complexes of pure-projective modules
  12. The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
  13. Sharp Li–Yau inequalities for Dunkl harmonic oscillators
  14. A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
  15. Free polynilpotent groups and the Magnus property
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