A hybrid Euler-Hadamard product and moments of \zeta'(\rho)

TL;DR

This paper uses a hybrid Euler-Hadamard product formula to derive conjectures about the moments of the derivative of the Riemann zeta-function at its zeros, integrating arithmetical factors naturally.

Contribution

It extends the hybrid formula approach to conjecture the moments of ta'( ho) over zeros, including arithmetical factors seamlessly.

Findings

Recovered a conjecture on moments of ta'( ho)

Integrated arithmetical factors naturally into the model

Connected random matrix theory with zeta function derivatives

Abstract

Keating and Snaith modeled the Riemann zeta-function \zeta(s) by characteristic polynomials of random NxN unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of \zeta(1/2+it) 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 \zeta(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 discrete moments of the derivative of the Riemann zeta-function averaged over the zeros of \zeta(s), incorporating the arithmetical factor in a natural way.