A NOTE ON AN 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$$\sum_{\rho}\frac{1}{|\frac{1}{2}-\rho|^4}=\frac{1}{2}\left(\frac{\xi''(\frac{1}{2})}{\xi(\frac{1}{2})}\right)^2-\frac{1}{6}\left(\frac{\xi^{(4)}(\frac{1}{2})}{\xi(\frac{1}{2})}\right) $$