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

TL;DR

This paper studies the positivity of the real part of the logarithmic derivative of the Riemann xi-function in the critical strip, providing bounds and examining implications of zeros off the critical line.

Contribution

It offers explicit bounds for the sum over zeros of the zeta function and explores the positivity condition assuming zeros off the critical line.

Findings

Explicit bounds for the sum over zeros of the zeta function.

Positivity of the real part of the logarithmic derivative in certain regions.

Implications of zeros off the critical line on the xi-function's properties.

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.