On the Hughes–Keating–O’Connell Conjecture: Quantified Negative Moment Bounds for ζ′(ρ) via Entropy–Sieve Methods Revisited

We study the negative discrete moments of the derivative of the Riemann zeta function at its nontrivial zeros, in connection with the Hughes--Keating--O’Connell conjecture. Building on the works of Gonek, Milinovich--Ng, Kirila, and the recent breakthrough of Bui--Florea--Milinovich, we introduce a new \emph{entropy--sieve method} (ESM). This framework combines short Dirichlet-polynomial approximations with entropy-based moment generating function bounds and a small-gap sieve, thereby controlling all appearances of $\zeta'(\rho)$ without assuming simplicity of zeros.Assuming the Riemann Hypothesis together with standard pair-correlation conjectures and a strengthened discrete moment hypothesis, we prove the quantified conditional bound \[ J_{-1}(T) \;=\; \sum_{0<\gamma\leq T} \frac{1}{|\zeta'(\tfrac12+i\gamma)|^{2}} \;\le\; C(\varepsilon)\, T (\log T)^{\varepsilon}, \qquad \text{for every fixed $\varepsilon>0$}, \] with an explicit dependence of the implicit constant on $\varepsilon$. This matches, up to logarithmic factors, the conjectured order $J_{-1}(T)\asymp T$ and improves on all previous conditional results.The analysis introduces several innovations: (i) a full cumulant control lemma for Dirichlet polynomials; (ii) explicit, non-circular parameter selection for approximation lengths and moments; and (iii) an entropy--sieve hybrid decay lemma that quantifies large-deviation probabilities for $\zeta'(\rho)$. Beyond the negative moment problem, the entropy--sieve framework illustrates the strength of entropy techniques in analytic number theory and points toward applications to $L$-functions and random matrix models.