If $L(\chi,1)=0$ then $\zeta(1/2+it)\neq0$

Abstract

Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ 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.