A new class of solutions to the van Dantzig problem, the Lee-Yang property, and the Riemann hypothesis

TL;DR

This paper explores the van Dantzig problem, linking it to the Lee-Yang property and the Riemann hypothesis, by identifying new classes of characteristic functions and analyzing their properties using probabilistic methods.

Contribution

It introduces a new class of entire functions in the set  that are not in _L, deepening the understanding of the van Dantzig problem and its connections to fundamental conjectures.

Findings

Identified closure properties of  and _L

Constructed new entire functions outside _L

Characterized van Dantzig random variables and their divisibility

Abstract

The purpose of this paper is to carry out an in-depth analysis of the intriguing van Dantzig problem which consists on characterizing the set $\mathbb{D}$ of analytic characteristic functions $\mathcal{F}$ which remains stable by the action of the mapping $V\mathcal{F}(t)=1/\mathcal{F}(it)$, $t\in\mathbb{R}$. % is also a characteristic function. We start by observing that the celebrated Lee-Yang property, appearing in statistical mechanics and quantum field theory, and the Riemann hypothesis can be both rephrased in terms of the van Dantzig problem, and, more specifically, in terms of the set $\mathbb{D}_L \subset \mathbb{D}$ of real-valued characteristic functions that belong to the Laguerre-P\'olya class. Motivated by these facts, we proceed by identifying several non-trivial closure properties of the set $\mathbb{D}$ and $\mathbb{D}_L$. This not only revisits but also, by means of probabilistic techniques, deepens the fascinating studies of the set of even characteristic functions in the Laguerre-P\'olya class carried out by P\'olya, de Bruijn, Lukacs, Newman and more recently by Newman and Wu, among others. We continue by providing a new class of entire functions that belong to the set $\mathbb{D}$ but not necessarily to $\mathbb{D}_L$, offering the first examples outside the set $\mathbb{D}_L$. This class, which is derived from some entire functions introduced by the second author, is in bijection with a subset of continuous negative-definite functions and includes several notable generalized hypergeometric-type functions. Besides identifying the characteristic functions, we also manage to characterize the pair of the corresponding van Dantzig random variables revealing that one of them is infinitely divisible. Finally, we investigate the possibility that the Riemann $\xi$ function belongs to this class.