An attempt of proof of Riemann Hypothesis

Abstract

This paper deals with an attempt of proof of the Riemann Hypothesis (RH). Let $T>10^{10}$ arbitrarily large. Let the region ${\Omega }_T=\{z=x+iy|\frac{1}{2}<x<1,0<y<T\}.$ 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 $\rho =\frac{1}{2}+\mathbf{u}+i\gamma$ whose real part is greater or equal to the real part of all the other roots in ${\Omega }_T$. Let $v\ge \frac{3}{2}$. Let $\epsilon >0$ arbitrarily small. We prove that $f(z)=\frac{{\zeta }^{\prime }(z)}{\zeta (z)}$ is analytic in the open disk ${\Omega }_{\epsilon }=\left\{\right|z-(\rho +\frac{\epsilon }{2}+v)\left|\right\}<v.$ Let $s=\rho +\epsilon$. We prove, from the Taylor series of $\zeta (s)$, that $f(s)\sim \frac{1}{\epsilon }\to \infty$ when $\epsilon \to 0$, and that, through the representation of $f(s)$ as a Taylor series, $f(s)=f(c_0)-(v-\frac{\varepsilon}{2})f'(c_0) +\frac{(v-\frac{\varepsilon}{2})^2}{2!}f''(c_0)-\frac{(v-\frac{\varepsilon}{2})^3}{3!}f^{(3)}(c_0)+\dots\mbox{\ for\ }c_0=ρ+\frac{\varepsilon}{2}+v,$ in ${\Omega }_{\epsilon }$, that $f(s)\nrightarrow \infty$ when $\epsilon \to 0$, a contradiction which allows us to prove RH.

This paper contains an irrecoverable error paragraph 6 page 4