On Hilbert's 8th Problem

A Hadamard factorisation of the Riemann $ξ$-function is used to characterise the zeros of the zeta function through the theory of generalized gamma convolutions (GGC). Riemann's reciprocal $ξ$-function is expressed, for $α>1$, 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 $α=\half$, where, through Riemann's original incomplete-gamma continuation of $ξ$, the primal law $ξ(\half+s)/ξ(\half)$ is shown to be a scale mixture of $Γ(2)$ densities. Two facts single out this representation: by the Steutel--Kristiansen theorem a positive mixture of $Γ(α)$ laws is infinitely divisible exactly when $α\in(0,2]$, so $Γ(2)$ 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.

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