Analytic Reformulations, Mellin Moments, Bose–Einstein Variance and Gamma–Zipf Mixtures for a Riemann Zeta Inequality

Abstract

We develop an integrated analytic and probabilistic framework for a proved real-variable inequality involving three consecutive shifts of the Riemann zeta function. After a change of parameter, the inequality is first placed in a chain of exact formulations: a weighted consecutive-zeta-ratio inequality, a strict three-moment inequality for the Bose–Einstein Mellin kernel, a variance-to-mean domination law for the normalized kernel, and a unit-step decrease condition for a zeta-ratio function. The Mellin formulation then becomes the gateway to genuinely new probabilistic consequences. We prove a Gamma–Zipf mixture representation, derive a strict inverse-moment inequality for the Zipf law, determine all positive integer moments, obtain an exact Hurwitz-zeta Laplace transform, and quantify the large-parameter variance excess. A Zipf differentiation identity further converts the stronger differential condition into a covariance inequality. The resulting framework separates exact equivalences from stronger sufficient conditions and shows how a zeta inequality generates nontrivial moment, mixture, and probabilistic structure.