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

Abstract

In this paper we show that for every Dirichlet $L$-function $L(s,\chi )$ and every $N\ge 2$ 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 $N=3$.

7 pages, 1 figure