Work

Zeros of L-functions and large partial sums of Dirichlet coefficients

Year: 2026

Type: article

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.

Source: Mathematische Annalen

DOI: 10.1007/s00208-026-03542-1

Authors: Bryce Kerr, Oleksiy Klurman, Jesse Thorner

Institutions: Australian Defence Force Academy, UNSW Canberra, University of Bristol, University of Illinois Urbana-Champaign

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: National Science Foundation, Simons Foundation, Australian Research Council