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

Abstract

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