Zeros of L-functions and large partial sums of Dirichlet coefficients
Abstract Let $$L(s,\pi )=\sum _{n=1}^{\infty }\lambda _{\pi }(n)n^{-s}$$ L ( s , π ) = ∑ n = 1 ∞ λ π ( n ) n - s be an L -function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $$\lambda _{\pi }(n)$$ λ π ( n ) strongly repel the low-lying zeros of $$L(s,\pi )$$ L ( s , π ) away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.
Authors
- Oleksiy Klurman
- Jesse Thorner (ORCID: https://orcid.org/0000-0003-0623-3473)
- Bryce Kerr (ORCID: https://orcid.org/0000-0002-8829-372X)
Institutions
- University of Illinois Urbana-Champaign (US)
- UNSW Sydney (AU)
- University of Bristol (GB)
- Australian Defence Force Academy (AU)
- UNSW Canberra
Publication Details
- Journal
- Mathematische Annalen
- Published
- 2026-08-28
- DOI
- https://doi.org/10.1007/s00208-026-03542-1
- Primary Topic
- Analytic Number Theory Research
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- National Science Foundation
- Simons Foundation
- Australian Research Council