On the Hughes--Keating--O’Connell Conjecture: \\ Quantified Negative Moment Bounds for ' () via Entropy--Sieve Methods Revisited

We study negative discrete moments of the derivative of the Riemann zeta function at its nontrivial zeros. Using a novel entropy--sieve method (ESM), and assuming the Riemann Hypothesis together with mild pair-correlation and discrete moment hypotheses, we establish a quantified conditional bound on negative moments: \ J-₁ (T) C () \, T (T) ^. \ Our approach combines Dirichlet polynomial approximations, Gaussian cumulant estimates, and a small-gap sieve. This framework matches the conjectured asymptotics up to logarithmic factors and has implications for the simplicity of zeros.