Leading-order complex moments of ζ′ at zeros

Determine, under the Riemann Hypothesis, the leading-order asymptotic for the complex k-th moment of the derivative of the Riemann zeta function at its non-trivial zeros: show that ∑_{0<γ≤T} [ζ′(1/2 + iγ)]^k ∼ [1/Γ(k+2)] ⋅ (T/2π) ⋅ (log(T/2π))^{k+1} as T → ∞ for Re(k) > −3.

Background

This conjecture states the leading-order asymptotics for complex moments of ζ′ at the non-trivial zeros, extending earlier work by the authors. In their companion paper, the authors conjectured this leading-order behavior using random matrix theory methods, and here they record it in a formal Conjecture environment.

Although the current paper’s methods focus on integer k, this conjecture asserts the asymptotic for complex k in the broader range Re(k) > −3.