Study and Consequences of the Sghiar's Function $\Circleds$ on the Riemann Hypothesis

Abstract

I study the Sghiar's function $\circledS$: $(X, z) \longmapsto \prod_{p \in \mathcal{P}} \frac{1}{1 - \frac{X}{p^z}}$, $\mathcal P $ the set of prime numbers. which is an extension of the Riemann zeta function [1-7], and I show that : $\zeta(s) = 0 \; \textit{and} \; \Re(s) > \frac{1}{2} \Rightarrow \circledS = 0$. We deduce the proof of the Riemann Hypothesis.

Keywords: Prime Number, Holomorphic function, the Riemann hypothesis.