Zeros of L(s)+L(2s)+⋯ + 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(Ns,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 that the method doesn't work in general for N=3.