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

Bryce Kerr; Oleksiy Klurman; Jesse Thorner

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arXiv · 2408.03938

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

Bryce Kerr·Oleksiy Klurman·Jesse Thorner

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Abstract

Let $L(s,\pi)=\sum_{n=1}^{\infty}\lambda_{\pi}(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $\lambda_{\pi}(n)$ strongly repel the low-lying zeros of $L(s,\pi)$ away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.

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Bryce Kerr, Oleksiy Klurman, Jesse Thorner. 2024-08-07. Zeros of $L$-functions and large partial sums of Dirichlet coefficients. https://arxiv.org/abs/2408.03938

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