[Moderator's Note: This was posted to the Forum by
Andrew Odlyzko *before* my note inviting him to do so!]
Apropos yesterday's posting by Arthur Smith and his exchanges with
Stevan Harnad concerning the Castro-Mahecha claim to have a proof
of the Riemann Hypothesis <
http://xxx.lanl.gov/abs/hep-th/0208221>,
which has caused a lot of discussion because of an article in
a Swedish newspaper and extensive slashdot discussions, let me
throw in a few comments.
1. I don't know of any experienced number theorist who has takes
this paper too seriously. The authors are not cranks, and there
may very well be more substance to their work than to that of
the Bogdanovs, but there are all sorts of warning signs that
persuaded me, as just one example, not to devote any real effort
to verifying this work. (And I should say that I am a bit of an
expert on the Riemann Hypothesis, and have been asked about the
Castro-Mahecha work by several editors and science writers in the
last couple of days.) First of all, the authors' record is not
all that great. A couple of years ago they were arguing that
the Riemann Hypothesis was likely false <
http://xxx.lanl.gov/abs/hep-th/0008055>
(although, to their credit, without claiming a rigorous proof),
and their previous attempt to prove the Riemann Hypothesis turned
out to have holes, as was pointed out in the paper they reference
as #19 (and the fact they do reference is again to their credit,
showing they are serious). More importantly, in skimming through
the paper, I find passages such as the following one towards the
bottom of page 6:
"..invariance ... follows by adopting a standard
regularization procedure of removing the infinities ..."
Well, essentially all the "standard regularization procedures"
that I am aware of are non-rigorous, so this alone makes
me very skeptical.
On balance I do not see anything here that would persuade
me to drop what I am doing and investigate this work in
detail, but I would change my work priorities if I heard of
some reputable mathematician taking this seriously.
2. Concerning Arthur's question of where the Castro-Mahecha
work fits in the mathematical literature, it is one of many
claimed "proofs" of the Riemann Hypothesis that are floating
around. A good reference for the more serious among them is
<
http://www.maths.ex.ac.uk/~mwatkins/zeta/RHproofs.htm>.
But there are still many more that don't make it online.
For example, every couple of months I get a new letter or
manuscript from a a senior (possibly retired) engineer
who fell in love with the subject. He has been doing
(rigorous and correct) numerical computations, it's just
that he has difficulty understanding the difference between
a pattern that is observed in the computed quantitities
and a rigorous proof.
The ease of electronic communication has certainly led to
a proliferation of unvetted materials getting widely distributed
(and sometimes even quoted by reputable news organizations).
3. The ease of electronic communication has also led to a
countervailing trend, namely the rise of new modes of communication
that allow us to cope with the increased load of fluff (something
that I wrote about in "The rapid evolution of scholarly communication,"
Learned Publishing, Jan. 2002, available at
<
http://www.dtc.umn.edu/~odlyzko/doc/eworld.html>). An outstanding
example of that is the Web site maintained by Matthew Watkins
listed above, part of an extensive collection of material about
the Riemann zeta function available at
<
http://www.maths.ex.ac.uk/~mwatkins/>.
It is not part of the traditional refereed literature, but it
is very widely used as a resource by number theory researchers.
(You might like to read about Watkins's background and interests
at <
http://www.maths.ex.ac.uk/~mwatkins/personal.htm>.)
Is the current situation any worse than it used to be? I don't
think so. We have always had incorrect papers slipping through
the refereeing process. Moreover, in addition to simple circulation
of crank materials, there were somewhat more ambiguous cases,
such as a journal that I encountered in a major university library
a year or two after getting my Ph.D. It was being published by
a couple of chaps from Argentina, and was devoted to "proofs of
the Riemann Hypothesis, Fermat's Last Theorem, and a couple of
other famous problems. As I recall, almost all of the "proofs"
were by the chaps publishing the journal. Well, it did not take
me too long to decide this was not serious, but I took some time
reading the journal, time that I would have saved had I had
access to more background material.
Andrew Odlyzko
P.S. I should also mention that the Castro-Mahecha paper has
been submitted to a conventional mathematics journal, and is
being refereed. The editors decided that it was not obvious
nonsense, and asked for an expert evaluation.
Received on Wed Mar 05 2003 - 13:44:06 GMT