{"id":340,"date":"2014-06-09T15:10:33","date_gmt":"2014-06-09T15:10:33","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/aslangul\/?p=340"},"modified":"2014-12-24T12:01:19","modified_gmt":"2014-12-24T12:01:19","slug":"conjecture-riemann","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/aslangul\/2014\/06\/09\/conjecture-riemann\/","title":{"rendered":"La conjecture de Riemann"},"content":{"rendered":"

La conjecture de Riemann r\u00e9siste depuis plus de 150 ans aux efforts des math\u00e9maticiens les plus imaginatifs. Riemann l’a \u00e9nonc\u00e9e en 1859 mais l’a laiss\u00e9e pour ainsi dire en plan avant de passer \u00e0 la suite de ce qui le pr\u00e9occupait sur l’instant. L’importance de cette conjecture r\u00e9side dans le fait qu’elle est au c\u0153ur de la probl\u00e9matique concernant la recherche de la r\u00e9partition si myst\u00e9rieuse des nombres premiers. De surcro\u00eet, bien des th\u00e9or\u00e8mes de la Th\u00e9orie des nombres reposent sur elle, et s’en trouveraient ipso facto<\/em> d\u00e9montr\u00e9s si elle l’\u00e9tait enfin. A n’en pas douter, cette conjecture, une fois prouv\u00e9e, figurerait dans la liste des th\u00e9or\u00e8mes utiles<\/em> au sens d\u00e9fini par Jean Dieudonn\u00e9 (ne pas oublier le pr\u00e9nom !) dans son tr\u00e8s beau livre “Pour l’honneur de l’esprit humain” <\/em>(Hachette, 1981), ce que confirme le fait qu’elle figure dans la fameuse liste de probl\u00e8mes dress\u00e9e par Hilbert en 1900, et qu’elle est aussi l’un des sept probl\u00e8mes du Prix du mill\u00e9naire selon le Clay Mathematical Institute<\/em>.<\/p>\n

Ce qu’a affirm\u00e9 Riemann concerne l’illustre fonction portant son nom, dont il pr\u00e9tend que les z\u00e9ros \"latex-image-1\"<\/a>, autres que les entiers pairs n\u00e9gatifs, sont tous<\/em> sur une droite du plan complexe situ\u00e9e \u00e0 la droite de l’axe imaginaire, \u00e0 la distance\u00a0 \"latex-image-2\"<\/a> de celui-ci. Depuis Riemann, des r\u00e9sultats importants ont \u00e9t\u00e9 \u00e9tablis ; par exemple, Hardy a d\u00e9montr\u00e9 en 1914 que la fameuse droite (dite ligne critique<\/em>) contient une infinit\u00e9 de z\u00e9ros, mais cela ne veut pas dire qu’ils y sont tous…<\/p>\n

\"L'annonce<\/a>
L’annonce du 1er avril 1997… \u00e0 la suite du canular de Enrico Bombieri.<\/figcaption><\/figure>\n

Il est int\u00e9ressant de constater que les tentatives de preuve ne sont pas l’apanage des math\u00e9maticiens : nombre de physiciens s’y sont essay\u00e9s sans y parvenir non plus malgr\u00e9 des approches sensiblement diff\u00e9rentes comme par exemple la recherche d’op\u00e9rateurs hermitiques dont le spectre serait pr\u00e9cis\u00e9ment constitu\u00e9 par tous les nombres \"iRkmoins12\"<\/a>…<\/p>\n

Le math\u00e9maticien britannique Matthew R. Watkins a constitu\u00e9 un site<\/a> rassemblant notamment diverses fausses preuves r\u00e9centes de cette conjecture, intitul\u00e9 “Proposed (dis)proofs of the Riemann Hypothesis”<\/em>, auquel le lecteur curieux est renvoy\u00e9. Il pourra notamment y d\u00e9couvrir la fameuse farce de Bombieri, un certain 1er avril (voir figure)…<\/p>\n

Sur ces questions, graves, profondes et parfois anecdotiques, on se doit de citer le tr\u00e8s beau livre de Marcus du Sautoy La symphonie des nombres premiers<\/em> (Collection Points – Science, Seuil 2007), dont la lecture est un voyage passionnant dans un univers \u00e0 la port\u00e9e de tous.<\/p>\n","protected":false},"excerpt":{"rendered":"

La conjecture de Riemann r\u00e9siste depuis plus de 150 ans aux efforts des math\u00e9maticiens les plus imaginatifs. Riemann l’a \u00e9nonc\u00e9e en 1859 mais l’a laiss\u00e9e pour ainsi dire en plan avant de passer \u00e0 la suite de ce qui le pr\u00e9occupait sur l’instant. L’importance de cette conjecture r\u00e9side dans le fait qu’elle est au c\u0153ur … Continuer la lecture de La conjecture de Riemann<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":347,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[6],"tags":[93,92,94,97,96,95],"class_list":["post-340","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-sciences","tag-conjecture-de-riemann","tag-farce-de-bombieri","tag-mathematiques","tag-riemann","tag-theoreme","tag-theorie-des-nombres"],"yoast_head":"\nLa conjecture de Riemann, par Claude Aslangul<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.futura-sciences.com\/aslangul\/2014\/06\/09\/conjecture-riemann\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La conjecture de Riemann, par Claude Aslangul\" \/>\n<meta property=\"og:description\" content=\"La conjecture de Riemann r\u00e9siste depuis plus de 150 ans aux efforts des math\u00e9maticiens les plus imaginatifs. 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