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

Abstract

The purpose of this paper to carry out an in-depth analysis of the intriguing van Dantzig problem. We start by observing that the celebrated Lee–Yang property and the Riemann hypothesis can be both rephrased in terms of this problem, and, more specifically, in terms of functions in the Laguerre–Pólya class. Motivated by these facts, we proceed by identifying several non-trivial closure properties enjoyed by the set of solutions to this problem. Not only does this revisit but also, by means of probabilistic techniques, deepens the fascinating and intensive studies of functions in the Laguerre–Pólya class. We continue by providing a new class of entire functions that are solutions to the van Dantzig problem. We also characterize the pair of the corresponding van Dantzig random variables. Finally, we investigate the possibility that the Riemann ξ function belongs to this class.

Locations

• arXiv (Cornell University) - View - PDF
• Annales de l’institut Fourier - View - PDF