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Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line

  • Jérémy Dousselin ORCID logo EMAIL logo
Veröffentlicht/Copyright: 27. Februar 2025
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 37 Heft 6

Abstract

Let 𝐹 be a linear combination (of a certain form) of N ≥ 1 Dirichlet 𝐿-functions attached to even (or odd) primitive characters. Selberg proved that a positive proportion of non-trivial zeros of 𝐹 lie on the critical line. Our work here is to provide an explicit lower bound for this proportion. In particular, we show that the lower bound 2.16 × 10 − 6 / ( N ⁢ log ⁡ N ) is admissible for large 𝑁.

Keywords: Dirichlet 𝐿-functions; critical line; mollification; linear combination of 𝐿 functions
MSC 2020: 11S40; 11M32; 11M26

Acknowledgements

I would like to thank my PhD advisor Youness Lamzouri for suggesting this problem to me, for his time and guidance, and for several improvement suggestions. Un remerciement à Pascal Ciot pour une aide bibliographique bien utile. A thanks to the anonymous referee for several useful comments and for the time spent meticulously proofreading this article.

  1. Communicated by: Guozhen Lu

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Received: 2023-12-11
Revised: 2023-06-08
Published Online: 2025-02-27
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
https://doi.org/10.1515/forum-2023-0453
Schlagwörter für diesen Artikel
Dirichlet 𝐿-functions; critical line; mollification; linear combination of 𝐿 functions

Artikel in diesem Heft

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
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