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 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 ' () 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) \;=\; ₀ ₓ 1|' (12+i) |^{2} \;\; C () \, T (T) ^, for every fixed 0, \ with an explicit dependence of the implicit constant on. This matches, up to logarithmic factors, the conjectured order J-₁ (T) 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 ' (). 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.
Zeraoulia Rafik (Mon,) studied this question.