  We point out an interesting occurrence of the sine kernel in connection with
the shifted moments of the Riemann zeta function along the critical line. We
discuss rigorous results in this direction for the shifted second moment and
for the shifted fourth moment. Furthermore, we conjecture that the sine kernel
also occurs in connection with the higher (even) shifted moments and show that
this conjecture is closely related to a recent conjecture by Conrey, Farmer,
Keating, Rubinstein, and Snaith.
