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

In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $\xi$-function in the region $1\mathrm{/}2+1\mathrm{/}\sqrt{\log t}<\sigma <1$, where $t$ is sufficiently large. We provide an explicit lower bound for $\mathfrak{R}{\sum }_{\rho }1\mathrm{/}(s-\rho )$, 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.