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

TL;DR

This paper demonstrates that large partial sums of Dirichlet coefficients of certain L-functions push their low-lying zeros away from the critical line, extending previous results with quantitative improvements.

Contribution

It provides a new, quantitative link between large partial sums of Dirichlet coefficients and the distribution of low-lying zeros of L-functions under a weak Ramanujan conjecture.

Findings

Large partial sums repel low-lying zeros from the critical line.

Quantitative bounds improve upon previous work.

Results apply to L-functions satisfying a weak Ramanujan conjecture.

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.