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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Riemann Hypothesis
Message-ID: <332@faron.UUCP>
Date: Thu, 22-Aug-85 11:21:11 EDT
Article-I.D.: faron.332
Posted: Thu Aug 22 11:21:11 1985
Date-Received: Sun, 25-Aug-85 01:25:21 EDT
Distribution: net
Organization: The MITRE Coporation, Bedford, MA
Lines: 33

The Riemann hypothesis proof by Matsumoto was withdrawn. Bombieri, Selberg,
and others found flaws in the proof that couldn't be patched.

For those of you who are not acquainted with Merten's conjecture: It was
recently disproved by some people at Bell Labs (Odlyzko and Lagarias I
believe). It states that:

	SUM (u(x))  <= sqrt(N)    where u(x) is the Mobius function
	x = 1,N			  defined as:

				1 if x is squarefree and has an even
				  number of distinct prime factors
				0 if x is not squarefree
				-1 if x is squarefree and has an odd 
				   number of distinct prime factors.
 
 
The truth of Merten's conjecture would have proved the Riemann Hypothesis.
 
An assertion was made recently that the proof of Mordell's conjecture
was 'close' to a proof of Fermat's Last Theorem. In fact, they are only
very vaguely related. The truth of Mordell's conjecture simply establishes
that for and GIVEN exponent N > 2: A^N + B^N = C^N has at most a finite
number of solutions. It doesn't say anywhere that the number of solutions
is zero.

A closer result is a recent one of Adelman et.al. who proved that
Fermat's Last Theorem was in fact true for an infinite number of exponents.
This of course does not mean that it's true for ALL exponents. As I recall
they did establish that the set of exponents for which the theorem was true
has positive density but that the density was less than 1.

Bob Silverman    (they call me Mr. 9)