Leading-order moments of mixed derivatives of ζ at zeros

Establish, under the Riemann Hypothesis, the leading-order asymptotic for mixed moments of derivatives of the Riemann zeta function at its non-trivial zeros: for integers n_1,…,n_k ≥ 0, show that ∑_{0<γ≤T} ζ^{(n_1)}(1/2 + iγ)⋯ζ^{(n_k)}(1/2 + iγ) ∼ (−1)^{n_1+⋯+n_k + k} ⋅ [n_1!⋯n_k!/(n_1+⋯+n_k + 1)!] ⋅ (T/2π) ⋅ (log(T/2π))^{n_1+⋯+n_k + 1} as T → ∞.

Background

This conjecture generalizes the k-th moment of ζ′ to products of higher derivatives with possibly distinct orders. It is obtained by differentiating the shifted product conjecture (RatioMom) with respect to the shifts and then specializing to zero.

The paper shows how to extract this leading term by expanding the integrand of the RatioMom conjecture and analyzing the contributions of the zero- and one-swap terms. The authors later confirm that the derived leading-order term matches the stated Conjecture.