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

Abstract

Let L(s,π)=∑n=1∞λπ(n)n-s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s,\pi )=\sum _{n=1}^{\infty }\lambda _{\pi }(n)n^{-s}$$\end{document} be an L-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of λπ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\pi }(n)$$\end{document} strongly repel the low-lying zeros of L(s,π)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s,\pi )$$\end{document} away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.