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Large sums of Hecke eigenvalues of holomorphic cusp forms

  • Youness Lamzouri EMAIL logo
Published/Copyright: November 3, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 2

Abstract

Let f be a Hecke cusp form of weight k for the full modular group, and let {λf⁢(n)}n≥1 be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of λf⁢(n), we investigate the range of x (in terms of k) for which there are cancellations in the sum Sf⁢(x)=∑n≤xλf⁢(n). We first show that Sf⁢(x)=o⁢(x⁢log⁡x) implies that λf⁢(n)<0 for some n≤x. We also prove that Sf⁢(x)=o⁢(x⁢log⁡x) in the range log⁡x/log⁡log⁡k→∞ assuming the Riemann hypothesis for L⁢(s,f), and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which Sf⁢(x)≫Ax⁢log⁡x, when x=(log⁡k)A. Our results are GL2 analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

Keywords: Hecke eigenvalues; sign changes; Sato–Tate measure; modular forms
MSC 2010: 11F30

Communicated by Jan Bruinier

Funding statement: The author is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

Acknowledgements

I would like to thank Emmanuel Kowalski for useful comments concerning the probabilistic random model for the Hecke eigenvalues in Section 3. I would also like to thank the anonymous referee for their comments and for suggesting a simpler proof of Theorem 1.1.

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Received: 2017-10-18
Revised: 2018-09-20
Published Online: 2018-11-03
Published in Print: 2019-03-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
  3. Infinite families of equivariantly formal toric orbifolds
  4. Bounds for GL3L-functions in depth aspect
  5. Nonlinear Dirichlet problems with unilateral growth on the reaction
  6. K-invariant cusp forms for reductive symmetric spaces of split rank one
  7. Cellularity in quotient spaces of topological groups
  8. Shifted convolution sums for higher rank groups
  9. Dimension quotients, Fox subgroups and limits of functors
  10. Large sums of Hecke eigenvalues of holomorphic cusp forms
  11. K-theory classification of graded ultramatricial algebras with involution
  12. Regularity of symbolic powers and arboricity of matroids
  13. On some problems concerning symmetrization operators
  14. An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
  15. Rankin–Selberg gamma factors of level zero representations of GLn
  16. Sobolev’s inequality for double phase functionals with variable exponents
  17. Independence of Artin L-functions
  18. Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
https://doi.org/10.1515/forum-2017-0222
Keywords for this article
Hecke eigenvalues; sign changes; Sato–Tate measure; modular forms

Articles in the same Issue

  1. Frontmatter
  2. Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
  3. Infinite families of equivariantly formal toric orbifolds
  4. Bounds for GL3L-functions in depth aspect
  5. Nonlinear Dirichlet problems with unilateral growth on the reaction
  6. K-invariant cusp forms for reductive symmetric spaces of split rank one
  7. Cellularity in quotient spaces of topological groups
  8. Shifted convolution sums for higher rank groups
  9. Dimension quotients, Fox subgroups and limits of functors
  10. Large sums of Hecke eigenvalues of holomorphic cusp forms
  11. K-theory classification of graded ultramatricial algebras with involution
  12. Regularity of symbolic powers and arboricity of matroids
  13. On some problems concerning symmetrization operators
  14. An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
  15. Rankin–Selberg gamma factors of level zero representations of GLn
  16. Sobolev’s inequality for double phase functionals with variable exponents
  17. Independence of Artin L-functions
  18. Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
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