An estimate of the second moment of a sampling of the Riemann zeta function on the critical line

We investigate the second moment of a random sampling $\zeta(1/2+iX_t)$ of the Riemann zeta function on the critical line. Our main result states that if $X_t$ is an increasing random sampling with gamma distribution, then for all sufficiently large $t$, \[\mathbb{E} |\zeta(1/2+iX_t)|^2 = \log t + O(\sqrt{\log t}\log\log t).\]