Full asymptotic for the second moment of ζ′ at zeros

Establish, under the Riemann Hypothesis, the full asymptotic expansion for the second moment of ζ′ at the non-trivial zeros: show that ∑_{0<γ≤T} [ζ′(1/2 + iγ)]^2 equals (1/2π) ∫_{1}^{T} [ (1/6)L^3 + (1/2)L^2(2γ_0 + A^{(0,0,1)}) + (1/2)L(−8γ_1 + 4γ_0 A^{(0,0,1)} + A^{(0,0,2)} + 2A^{(0,1,1)}) + (1/6)(−12γ_0^3 − 36γ_0γ_1 + 6γ_2 − 24γ_1 A^{(0,0,1)} + 6γ_0 A^{(0,0,2)} + A^{(0,0,3)} + 12γ_0 A^{(0,1,1)} + 3A^{(0,1,2)} − 3A^{(0,2,1)} + 6A^{(1,1,1)}) ] dt + O(T^{1/2+ε}), where L = log(t/(2π)), γ_m are the Stieltjes constants from the Laurent expansion of ζ(s) at s=1, and each A^{(i,j,k)} is an explicit arithmetic Euler-product coefficient obtained by differentiating the arithmetic factor A_{ {α,β} }(δ) with respect to α, β, and δ and then setting α=β=δ=0.

Background

This conjecture provides, for the first time, a full asymptotic expansion for the second moment of ζ′ at the zeros, including lower-order terms involving Stieltjes constants and derivatives of an arithmetic Euler product. It is derived by expanding the RatioMom integrand and computing the necessary derivatives of the arithmetic factor.

The authors support the conjecture with numerical evidence, computing A^{(i,j,k)} by expanding log A_{ {α,β} }(δ) and summing over primes (first 1000 primes in their experiments), and comparing the predicted integral polynomial with empirical sums over zeros.