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H 1 Remarks on the Riemann hypothesis

Auteurs : Newman, Charles (Auteur de la Conférence)
CIRM (Editeur )

Résumé : One fairly standard version of the Riemann Hypothesis (RH) is that a specific probability density on the real line has a moment generating function (Laplace transform) that as an analytic function on the complex plane has all its zeros pure imaginary. We’ll review a series of results that span the period from the 1920’s to 2018 concerning a perturbed version of the RH. In that perturbed version, due to Polya, the log of the probability density is modified by a kind of mass term (in quantum field theory language). This gives rise to an implicitly defined real constant known as the de Bruijn-Newman Constant, Lambda. The conjecture and now theorem (Newman 1976, Rodgers and Tau 2018) that Lambda is greater than or equal to zero is complementary to the RH which is equivalent to Lambda less than or equal to zero; The conjecture/theorem is a version of the dictum that the RH, if true, is only barely so. We’ll also briefly discuss some connections with quantum field theory and the Lee-Yang circle theorem.

Codes MSC :
11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
60K35 - Interacting random processes; statistical mechanics type models; percolation theory

 Informations sur la Vidéo Langue : Anglais Date de publication : 11/09/2018 Date de captation : 30/08/2018 Collection : History of Mathematics ; Probability and Statistics ; Number Theory Sous collection : Research talks Format : MP4 Domaine : History of Mathematics ; Probability & Statistics ; Number Theory Durée : 00:48:06 Audience : Doctorants , Post - Doctorants ; Chercheurs Download : https://videos.cirm-math.fr/2018-08-30_Newman.mp4 Informations sur la rencontre Nom de la rencontre : Advances in Statistical Mechanics / Avancées en mécanique statistiqueOrganisateurs de la rencontre : Arguin, Louis-Pierre ; Gayrard, Véronique ; Kistler, Nicola ; Kourkova, IrinaDates : 27/08/2018 - 31/08/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1855.htmlCitation Data DOI : 10.24350/CIRM.V.19441503 Cite this video as: Newman, Charles (2018). Remarks on the Riemann hypothesis. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19441503 URI : http://dx.doi.org/10.24350/CIRM.V.19441503

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