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

In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function for $1\mathrm{/}2<\sigma <1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\mathrm{\Re }{\sum }_{\rho }1\mathrm{/}(s-\rho )$, where the sum runs over the zeros of $\zeta (s)$ on the line $1\mathrm{/}2+it$. We also check the positivity of $\mathrm{\Re }{\xi }^{\mathrm{\prime }}\mathrm{/}\xi (s)$ for $1\mathrm{/}2<\sigma <1$ assuming that there occur a non-trivial zeros of $\zeta (s)$ off the critical line.