Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site faron.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!whuxlm!harpo!decvax!linus!faron!bs From: bs@faron.UUCP (Robert D. Silverman) Newsgroups: net.math Subject: Riemann Hypothesis Message-ID: <193@faron.UUCP> Date: Thu, 27-Dec-84 14:36:27 EST Article-I.D.: faron.193 Posted: Thu Dec 27 14:36:27 1984 Date-Received: Fri, 28-Dec-84 08:04:24 EST Organization: The MITRE Corp., Bedford, Ma. Lines: 27 For those of you who haven't heard yet, the Riemann Hypothesis has apparantly been proved by a mathematical physicist at the University of Paris. His name is Matsumoto (sp?). The proof has been seen by others and appears to be correct. It is also likely that the Generalized Riemann Hypothesis has been proved along with it. For those of you who are unfamiliar with the problem it is considered to be (or was) the most outstanding problem in all of mathematics. The Zeta function is an analytic function of z = a+bi in the half plane a > 1 and can be extended so that it is analytic everywhere except at z = 1 where it has a first order pole. The hypothesis states that all of its zeros lie on the line Real(z) = 1/2 and has many implications in the theory of prime numbers. A good reference for the problem may be found in (among others) "Topics from the Theory of Numbers", Emil Grosswald, MacMillan Co., 1966. One very important aspect of the problem is that if the generalized hypothesis it true there exists a deterministic, polynomial time algorithm for proving numbers prime. I do not know the details of the proof except in a very vague way. Apparantly there is a class of operators relating to the Zeta function whose positive definiteness is equivalent to the hypothesis. Matsumoto has shown that those operators are in fact positive definite.