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 ζ ( 1 / 2 + i X 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, E | ζ ( 1 / 2 + i X t ) | 2 = log t + O ( log t log log t ) .