On the joint second moment of zeta and its logarithmic derivative

Assuming the Riemann Hypothesis, Goldston, Gonek and Montgomery (2001) 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 $|\zeta (\frac{1}{2}+it)|^2$. 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.