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, χ) and every N ≥ 2 the Dirichlet series L(s, χ) + L(2s, χ) + • • • + L(N s, χ) have infinitely many zeros for σ > 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.