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

Abstract

Let $L(s,\pi )={\sum }_{n=1}^{\infty }{\lambda }_{\pi }(n)n^{-s}$ 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.

23 pages