Note on the positivity of the real part of the log-derivative of the Riemann $\xi$-function near the critical line

Abstract

In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $\xi$-function in the region $1/2+1/\sqrt{\log t}<\sigma<1$, where $t$ is sufficiently large. We provide an explicit lower bound for $\mathfrak{R}\sum_{\rho}1/(s-\rho)$, where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the log-derivative of the Riemann $\xi$-function in the provided region, assuming that there are non-trivial zeta-zeros off the critical line.