On the Hughes–Keating–O’Connell Conjecture: Entropy-Sieve Methods for Negative Moments of ζ′(ρ)

We investigate the negative discrete moments of the derivative of the Riemann zeta function at its nontrivial zeros, focusing on the Hughes–Keating–O’Connell conjecture. Building on the earlier frameworks of Gonek, Milinovich–Ng, Kirila, and the recent breakthrough of Bui–Florea–Milinovich, we introduce a hybrid entropy–sieve method (ESM). This method refines Dirichlet-polynomial approximations by quantifying entropy of local distributions of \( D_X(\gamma) \) and controlling contributions from both small gaps and low-entropy blocks. Assuming the Riemann Hypothesis and standard pair-correlation conjectures, we prove the near-optimal conditional upper bound \( J_{-1}(T) \;=\; \sum_{0<\gamma\leq T} \frac{1}{|\zeta'(\rho)|^{2}} \;\ll\; T(\log T)^{\varepsilon}. \)This matches, up to logarithmic factors, the conjectured order \( J_{-1}(T)\asymp T \), improving upon previous conditional bounds in the literature. Our approach complements the sieve and moment methods of Bui–Florea–Milinovich and the entropy-based large deviation heuristics of Harper, while introducing new tools such as a uniform Dirichlet-polynomial approximation with explicit coefficients and quantitative entropy-decay estimates. Beyond these results, the ESM framework highlights the utility of entropy techniques in analytic number theory, suggesting applications to related problems in L-function theory and random matrix models.