  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.

Number Theory;Complex Variables