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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.