The Riemann hypothesis via the Mellin transform, power series and the reflection relations

Abstract: A proof of the Riemann hypothesis is proposed by relying on the properties of the Mellin transform. The function $\mathfrak{G}{\eta}\left(t\right)$ is defined on the set $\bar{\mathbb{R}}+$ of the non-negative real numbers, in term of a special power series, in such a way that the Mellin transform $\hat{\mathfrak{G}}{\eta}\left(s\right)$ of the function $\mathfrak{G}{\eta}\left(t\right)$ does not vanish in the fundamental strip $0<\operatorname{Re} s <1/2$. In this strip every zero of the Riemann zeta function $\zeta\left(1-s\right)$ is a zero of the function $\hat{\mathfrak{G}}_{\eta}\left(s\right)$. Consequently, it is proved that no zero of the Riemann zeta function $\zeta\left(s\right)$ exists in the strip $1/2<\operatorname{Re} s <1$. The reflection relations, which hold around the line $\operatorname{Re} s =1/2$ for $s\neq 0,1$, prove that no zero of the Riemann zeta function $\zeta\left(s\right)$ exists in the strip $0<\operatorname{Re} s<1/2$. In conclusion, it is proved that no zero of the Riemann zeta function $\zeta\left(s\right)$ exists in the strip $0<\operatorname{Re} s<1$ for $\operatorname{Re} s\neq 1/2$.