Sacha Grover

Higher order derivative moments of CUE characteristic polynomials and the Riemann zeta function

We use Random matrix theory for the Circular Unitary Ensemble (CUE) to study integer moments of derivatives of the Riemann zeta function off the critical line. We investigate mixed integer moments of derivatives of characteristic polynomials in the CUE to provide an expression in the large N limit at fixed points inside the unit disc. The moments are expressed as an explicit combinatorial sum involving hypergeometric functions. This method allows us to recover the well-established result of Diaconis-Gamburd in the limit z->0, as well as a generalisation for arbitrary 0<=|z|<1 of some results due to Simm-Wei. Of particular interest is the leading asymptotic as z->1, which can be directly related to the analogous moment of the Riemann zeta function as one approaches the half-line. We then demonstrate that moments of the Riemann zeta function give rise to the same combinatorial sum as observed in the CUE. This is shown unconditionally for sufficiently low-order moments and, assuming the Lindelöf hypothesis, is proven in full generality.