Shifted discrete zeta moments at zeros via the Ratios Conjecture

Establish, under the Riemann Hypothesis and small-shift conditions (Re(α_j) with |Re(α_j)| < 1/4 and |Im(α_j)| ≪ε T^{1−ε} for every ε > 0, and |δ| < 1/4), the full asymptotic expansion for the sum S(T) = ∑_{0<γ≤T} ζ(1/2 + iγ + α_1)⋯ζ(1/2 + iγ + α_k). Specifically, show that S(T) equals the derivative with respect to δ at δ = 0 of (1/2π) ∫_{1}^{T} [Z_{α_1,…,α_k,δ}(t) + Σ_{j=1}^k (t/(2π))^{−α_j−δ} Z_{α_1,…,α_{j−1},−δ,α_{j+1},…,α_k,−α_j}(t)] dt, plus the term (T/2π) log(T/2π), with an error term O(T^{1/2+ε}). Here Z_{α_1,…,α_k,δ} := [∏_{j=1}^k ζ(1+α_j+δ)/ζ(1+α_j)] ⋅ A_{ {α_1,…,α_k} }(δ), and the arithmetic factor A_{ {α_1,…,α_k} }(δ) is defined by the Euler product over primes of [1 + Σ_{m=1}^k F_m(p)] / ∏_{j=1}^k (1 − p^{−(1+α_j)}), with F_m(p) = (−1)^m Σ_{J⊂{1,…,k}, |J|=m} p^{−(m + (m−1)δ + Σ_{j∈J} α_j)}.

Background

This is the central conjecture of the paper, derived using the Ratios Conjecture methodology. It provides a full asymptotic expansion for products of shifted zeta values evaluated at the non-trivial zeros, expressed as a δ-derivative of an explicit integral whose integrand contains the zero-swap and one-swap contributions. The term Z_{α1,…,α_k,δ} bundles zeta factors and an arithmetic Euler product A{ {α_1,…,α_k} }(δ).

The conjecture assumes the Riemann Hypothesis and small-shift conditions. The error term O(T^{1/2+ε}) follows the original statements of the recipe/Ratios Conjecture framework. Proving this conjecture would rigorously validate the Ratios Conjecture prediction for these shifted discrete moments and enable full asymptotic expansions for moments of mixed derivatives via differentiation with respect to the shifts.