A hybrid Euler–Hadamard product and moments of ζ'(ρ)

Abstract

Abstract Keating and Snaith modeled the Riemann zeta-function ζ(s) by characteristic polynomials of random N×N unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of ζ(1/2+i t) when k > -1/2. However, an arithmetical factor, widely believed to be part of the leading term coefficient, had to be inserted in an ad hoc manner. Gonek, Hughes and Keating later developed a hybrid formula for ζ(s) that combines a truncation of its Euler product with a product over its zeros. Using it, they recovered the moment conjecture of Keating and Snaith in a way that naturally includes the arithmetical factor. Here we use the hybrid formula to recover a conjecture of Hughes, Keating and O'Connell concerning the discrete moments of the derivative of the Riemann zeta-function averaged over the zeros of ζ(s), incorporating the arithmetical factor in a natural way.