A note on series equivalent of the Riemann hypothesis

Abstract

In this manuscript we denote by $\sum _{\rho }$ a sum over the non trivial zeros of Riemann zeta function (or over the zeros of Riemann’s xi function), where the zeros of multiplicity k are counted k times. We prove a result that the Riemann Hypothesis is true if and only if

$$\begin{aligned} \sum _{\rho }\frac{1}{|\frac{1}{2}-\rho |^2}=\frac{\xi ''(\frac{1}{2})}{\xi (\frac{1}{2})} \end{aligned}$$