If $L(\chi ,1)=0$ then $\zeta (1/2+it)\ne 0$

Abstract

Let $L(s)={\sum }_{n=1}^{+\infty }\frac{a(n)}{n^s}$ be a Dirichlet series were $a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$ extends to a holomorphic function on the open half space $\Re s>1-\delta$, $\delta >0$ and $L(1)=0$ then such a half space is a zero free region of the Riemann zeta function $\zeta (s)$. Similar results is proven for completely multiplicative functions defined on the space of the ideals of the ring of the algebraic integers of a number field of finite degree.

24 pages. Published in Riv. Mat. Univ. Parma with the title "Non vanishing of Dirichlet series of completely multiplicative functions"