This paper deals with an attempt of proof of the Riemann Hypothesis (RH). Let $T>10^{ 10}$ arbitrarily large. Let the region $\Omega_T=\Big\{ z=x+i y\ \Big|\ \frac{ 1}{ 2}<x<1, \ 0<y<T\Big\}.$ There is a finite number $N_T$ of roots of $\zeta(z)$ in $\Omega_T$. The aim of the paper is to prove that $N_T=0$. Suppose that $N_T>0$. There exists at least one root $\rho=\frac{ 1}{ 2}+{ \bf u}+i\gamma $ whose real part is greater or equal to the real part of all the other roots in $\Omega_T$. Let $v\geq \frac{ 3}{ 2}$. Let $\varepsilon>0$ arbitrarily small. We prove that $f(z)=\frac{ \zeta'(z)}{ \zeta(z)}$ is analytic in the open disk $\Omega_\varepsilon=\Big\{ \Big|z-\Big(\rho+\frac{ \varepsilon}{ 2}+v\Big)\Big|\Big\}< v.$ Let $s=\rho+\varepsilon$. We prove, from the Taylor series of $\zeta(s)$, that $f(s)\sim \frac{ 1}{ \varepsilon}\rightarrow \infty$ when $\varepsilon\rightarrow 0$, and that, through the representation of $f(s)$ as a Taylor series, $f(s)=f(c_0)-(v-\frac{ \varepsilon}{ 2})f'(c_0) +\frac{ (v-\frac{ \varepsilon}{ 2})^2}{ 2!}f''(c_0)-\frac{ (v-\frac{ \varepsilon}{ 2})^3}{ 3!}f^{ (3)}(c_0)+\dots\mbox{ \ for\ }c_0=\rho+\frac{ \varepsilon}{ 2}+v,$ in $\Omega_\varepsilon$, that $f(s)\not\rightarrow \infty$ when $\varepsilon\rightarrow 0$, a contradiction which allows us to prove RH.
Categories: math.GM