On Hilbert's 8th Problem

Abstract

A Hadamard factorisation of the Riemann $\xi$-function is used to characterise the zeros of the zeta function through the theory of generalized gamma convolutions (GGC). Riemann's reciprocal $\xi$-function is expressed, for $\alpha>1$, as the Laplace transform of a GGC, with an explicit Levy form and a reciprocal Thorin measure given by a sine-squared transform. The construction is then carried to the centre $\alpha=\half$, where, through Riemann's original incomplete-gamma continuation of $\xi$, the primal law $\xi(\half+s)/\xi(\half)$ is shown to be a scale mixture of $\Gamma(2)$ densities. Two facts single out this representation: by the Steutel--Kristiansen theorem a positive mixture of $\Gamma(\alpha)$ laws is infinitely divisible exactly when $\alpha\in(0,2]$, so $\Gamma(2)$ is the threshold shape at which positivity of the mixing measure alone secures infinite divisibility; and that mixing measure $M$ is positive, which we prove cell by cell through an average-of-powers identity that turns an apparently signed alternating series into a monotone Leibniz series. The positive $M$ is the input the Thorin/GGC condition requires.