On Hilbert's 8th Problem

Abstract

A Hadamard factorisation of the Riemann $ξ$-function is constructed to characterise the zeros of the Riemann zeta function. The argument proceeds via a probabilistic reformulation: the Riemann Hypothesis is shown to be equivalent to Thorin's condition, namely that $ξ(\half)/ξ(\half+\sqrt s\,)$ is the Laplace transform of a generalised gamma convolution (GGC). A GGC random variable realising this Laplace transform is constructed unconditionally, using Lévy representations of the Gamma function, the Euler product for the zeta function, and Ramanujan's Master Theorem, thereby establishing the hypothesis.