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

Abstract

The Hadamard-Weierstrass factorisation of an entire function in the Laguerre-Pólya class is dual to a pair of probabilistic objects: a van Dantzig pair of characteristic functions and a Wald couple of infinitely divisible random variables. The reciprocal of such a function is the Laplace transform of a generalised gamma convolution (GGC) whose Thorin measure encodes the zeros, and Thorin's condition, a real Laplace identity on $(0,\infty )$, is equivalent to the absence of zeros off the critical axis. We prove this duality, together with a closed-form formula for the Thorin density in terms of any Lévy representation. The framework is then applied to the Gamma, hyperbolic, Bessel and Macdonald functions, to the Riemann $\xi$-function, to Dirichlet and modular $L$-functions, and to Dedekind's $\eta$. For Ramanujan's $\tau$ it is set up but not carried through, since reality of the zeros of the associated $L$-function is open. For $\xi$ we prove, unconditionally, that the heat trace $W(t)=\frac{1}{2}{\sum }_{\rho }\exp \{(\rho -\frac{1}{2}{)}^{2}t\}$ is positive and strictly decreasing on $(0,\infty )$, so that $\xi (\frac{1}{2})/\xi (\frac{1}{2}+\sqrt{u})$ is the Laplace transform of a self-decomposable law, and that $\xi (\alpha )/\xi (\alpha +\sqrt{s})$ is the Laplace transform of a GGC for every $\alpha$ exceeding the real part of every non-trivial zero, hence unconditionally for every $\alpha \ge 1$. Complete monotonicity of $W$, equivalently the GGC property at the centre, is shown to be equivalent to the Riemann hypothesis. Self-decomposability is therefore the unconditional ceiling.

References in corpus (5)

• Zeros of some special entire functions
• Further examples of GGC and HCM densities
• A probabilistic interpretation of a sequence related to Narayana polynomials
• On Hilbert's 8th Problem
• Wronskians of Fourier and Laplace Transforms