The number of zeros of linear combinations of $L$-functions near the critical line

Abstract

In this paper, we investigate the zeros near the critical line of linear combinations of $L$-functions belonging to a large class, which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$. More precisely, if $L_1,\dots ,L_J$ are distinct primitive $L$-functions with $J\ge 2$, and $b_j$ are any nonzero real numbers, we prove that the number of zeros of $F(s)={\sum }_{j\le J}{b}_j{L}_j(s)$ in the region $\text{Re}(s)\ge 1/2+1/G(T)$ and $\text{Im}(s)\in [T,2T]$ is asymptotic to $K_{0}TG(T)/\sqrt{\log G(T)}$ uniformly in the range $\log \log T\le G(T)\le (\log T{)}^{\nu }$, where $K_0$ is a certain positive constant that depends on $J$ and the $L_j$'s. This establishes a generalization of a conjecture of Hejhal in this range. Moreover, the exponent $\nu$ verifies $\nu \asymp 1/J$ as $J$ grows.

arXiv commentMinor corrections. 55 pages. To appear in Journal d'Analyse Mathématique