On Hilbert's 8th Problem

Abstract

A Hadamard factorisation of the Riemann $\hat{I}¾$-function is used to characterise the zeros of the zeta function through the theory of generalized gamma convolutions (GGC). Riemann's reciprocal $\hat{I}¾$-function is expressed, for $\hat{I}\pm >1$, 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 $\hat{I}¾$, the primal law $ξ(\half+s)/ξ(\half)$ is shown to be a scale mixture of $\hat{I}“(2)$ densities. Two facts single out this representation: by the Steutel--Kristiansen theorem a positive mixture of $\hat{I}“(\hat{I}\pm )$ laws is infinitely divisible exactly when $\hat{I}\pm \in (0,2]$, so $\hat{I}“(2)$ 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.

Cited by in corpus (1)

• Riemann, Thorin, van Dantzig Pairs, Wald Couples and Hadamard Factorisation