A new class of solutions to the van Dantzig problem, the Lee–Yang property, and the Riemann hypothesis
Annales de l'Institut Fourier, Online first, 45 p.

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.

Dans cet article, nous effectuons une analyse approfondie du curieux problème de van Dantzig. Nous commençons par observer que la propriété célèbre de Lee–Yang et l’hypothèse de Riemann peuvent être reformulées en termes de ce problème, et, plus spécifiquement, en termes de fonctions dans la classe de Laguerre–Pólya. Motivés par ces faits, nous poursuivons en identifiant plusieurs propriétés non triviales satisfaites par l’ensemble des solutions à ce problème. Cela nous permet non seulement de revisiter mais aussi, au moyen de techniques probabilistes, d’approfondir les nombreuses études fascinantes développées pour les fonctions dans la classe de Laguerre–Pólya. Nous continuons en caractérisant une nouvelle classe de fonctions entières qui sont solutions du problème de van Dantzig. Nous donnons également la paire de variables aléatoires de van Danzig correspondante. Enfin, nous étudions la possibilité que la fonction ξ de Riemann appartienne à cette classe.

Online First:
DOI: 10.5802/aif.3600
Classification: 42A38, 33C47, 47D07, 30C15
Keywords: van Dantzig problem, Characteristic functions, Laguerre–Pólya entire functions, Generalized hypergeometric functions, Lee–Yang property, Riemann hypothesis, Self-similar Markov processes
Mot clés : Problème de van dantzig, Fonctions charactéristiques, fonctions entieres de Laguerre–Pólya, fonctions hypergeométriques généralisées, propriété de Lee–Yang, hypothese de Riemann, processus de Markov auto-similaires
Konstantopoulos, Takis 1; Patie, Pierre 2; Sarkar, Rohan 3

1 Department of Mathematical Sciences University of Liverpool Liverpool, L69 7ZL (UK)
2 School of Operations Research and Information Engineering, Cornell University Ithaca, NY 14853 (USA)
3 Department of Mathematics University of Connecticut Storrs, CT 06269 (USA)
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Konstantopoulos, Takis; Patie, Pierre; Sarkar, Rohan. A new class of solutions to the van Dantzig problem, the Lee–Yang property, and the Riemann hypothesis. Annales de l'Institut Fourier, Online first, 45 p.

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