On a positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function

Abstract

In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function for $1/2<\sigma<1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\Re\sum_{\rho} 1/(s-\rho)$, where the sum runs over the zeros of $\zeta(s)$ on the line $1/2+it$. We also check the positivity of $\Re \xi'/\xi(s)$ for $1/2<\sigma<1$ assuming that there occur a non-trivial zeros of $\zeta(s)$ off the critical line.