On the joint second moment of zeta and its logarithmic derivative

Assuming the Riemann Hypothesis, Goldston, Gonek and Montgomery \cite{GGM} studied the second moment of the log-derivative of $\zeta$, shifted away from the half line by $a/\log T$, and its connection with the pair correlation conjecture. In this paper, we consider a weighted version of this problem, where the average is tilted by $|\zeta(\frac{1}{2}+it)|^2$. More precisely, we provide an upper and a lower bound for the second moment of zeta times its logarithmic derivative, $a/\log T$ away from the critical line.