Leading-order kth moment of the nth derivative of ζ at zeros

Establish, under the Riemann Hypothesis, the leading-order asymptotic for the k-th moment of the n-th derivative of the Riemann zeta function at its non-trivial zeros: for non-negative integer k and n ∈ ℕ, show that ∑_{0<γ≤T} [ζ^{(n)}(1/2 + iγ)]^{k} ∼ (−1)^{k(n+1)} ⋅ [(n!)^k / (kn + 1)!] ⋅ (T/2π) ⋅ (log(T/2π))^{kn + 1} as T → ∞.

Background

This conjecture is the natural specialization of the mixed-derivative leading-order conjecture when all derivative orders are equal. It provides an explicit factorial coefficient and log-power consistent with predictions from the Ratios Conjecture and random matrix analogues.

The authors derive this by applying the RatioMom framework and then equating all derivative orders, confirming the structure of the leading term via symmetric polynomial identities.