The Riemann Zeta - Function and the Sine Kernel

Abstract. 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 establish this occurrence rigorously for the shifted second moment and, under some constraints on the shifts, for the shifted fourth moment. Our proofs of these results closely follow the classical proofs for the non-shifted moments of the Riemann zeta-function. 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 [CFKRS1, CFKRS2].