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

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 $\Re {\xi }^{\prime }/\xi (s)$ for $1/2<\sigma <1$ assuming that there occur a non-trivial zeros of $\zeta (s)$ off the critical line.

Cited by in corpus (1)

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