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

TL;DR

This paper explores the connection between zeros of entire functions and probabilistic methods, demonstrating this duality on important functions like the Riemann zeta and L-functions.

Contribution

It introduces a novel duality linking Hadamard factorisation with van Dantzig pairs and Wald couples, applying it to key special functions.

Findings

Established a duality between Hadamard factorisation and probabilistic pairs.

Applied the methodology to Riemann zeta, xi-functions, and L-functions.

Demonstrated the approach on classical special functions.

Abstract

Zeros of entire functions can be found using either the Fourier methods of Riemann-Polya or the Generalized Gamma Convolution (GGC) methods of Thorin-Bondesson. This connection is based on a duality between the Hadamard-Weierstrass factorisation and van Dantzig Pairs-Wald couples of random variables. We demonstrate the methodology on particular functions including the Riemann zeta and xi-functions, Ramanujan's tau function, L-functions and Gamma and Hyperbolic functions.