Zeros of $L(s)+L(2s)+\cdots+L(Ns)$ in the region of absolute convergence

TL;DR

This paper investigates the zeros of sums of Dirichlet $L$-functions evaluated at multiples of $s$, proving infinite zeros in the region of absolute convergence for many cases, with some limitations demonstrated through examples.

Contribution

It establishes the existence of infinitely many zeros for sums of Dirichlet $L$-functions in the region $ ext{Re}(s)>1$, extending understanding of their zero distribution.

Findings

Infinite zeros for sums of Dirichlet $L$-functions when $N eq 3$

Zeros exist for many $L$-functions with Euler products for large $N$ or $N=2$

Method does not extend to $N=3$ in general

Abstract

In this paper we show that for every Dirichlet $L$-function $L(s,\chi)$ and every $N\geq 2$ the Dirichlet series $L(s,\chi)+L(2s,\chi)+\cdots+L(Ns,\chi)$ have infinitely many zeros for $\sigma>1$. Moreover we show that for many general $L$-functions with an Euler product the same holds if $N$ is sufficiently large, or if $N=2$. On the other hand we show with an example the the method doesn't work in general for $N=3$.