[FOM] on harvey friedman's message, "Re on harvey friedman's 'number theorists'" of April 7.
gstolzen at math.bu.edu
Sun Apr 9 18:04:38 EDT 2006
Harvey begins by quoting from my message "on harvey friedman's
'number theorists'" of April 4th.
> > I'm surprised that you don't tell us how this interest is manifested
> > mathematically. Isn't that important? I'd like to see some of the
> > work that he did on questions of this kind.
> He is one of the three people I contacted, and I haven't yet heard
> from from any three. I have to admit that I am becoming pessimistic
> about hearing from them.
GS: Don't be. I don't think not hearing back for a while should count
for anything. There are things we know in our bones re mathematics but
find it difficult to articulate. This might well be one of them.
> My impression is that most of the leading senior number theorists
> have published bounds either improving previous bounds or establishing
> a bound where no bound previously existed.
GS: I would be astonished if this were not true. But, for me, the
questions are (1) what kind of bounds (e.g., improving a 3 to a 2),
(2) how much, if any, of it was done for its "intrinsic interest"
and (3) what other reasons were given?
I was just reviewing what used to be the most famous case of this,
a sign change for pi - li. The classical existence theorem was proved
by Littlewood in 1914. Then, in 1933, Skewes (who apparently was a
graduate student) worked out a humongous bound on the assumption that
the Riemann hypothesis is true. Finally, in 1955, he got a second
bound on the assumption that it is false and took the max of the two.
GS: So far as I can tell, the only reason that number theorists were
interested in Skewes' number was that the nature of Littlewood's
argument (by cases, depending on whether RH is true or false) made it
seem "intrinsically nonconstructive." Finally, in 1966, by a different
method, Sherman Lehman got a much better bound. But he was playing a
very different game.
GS: Why was Littlewood's theorem interesting to people? As Ingham
explains, according to the best table of values then available, pi(n)
was always < li(n), even though the ratio goes to 1 pretty quickly.
But, so the thinking went, if this striking relationship were to fail
for larger n, then, by analogy, the fact that the Riemann hypothesis
had also been confirmed up to a high value would no longer carry the
same evidentiary weight that it does.
> The mere existence of a an effective bound for a Pi03 is of
> intrinsic interest, and the interest to number theorists increases
> as the bounds get lower and lower.
GS: We still seem to disagree about this. However, I'm still not
clear about what you mean by "intrinsic interest." (It would help if
you quantified it.) As I've said before, if it includes fascination
and we're talking about the absence of such a bound rather than the
construction of one, then, in some cases. sure.
GS: As for an apparent interest in the construction of a bound, I
think Skewes' number is hard to beat. It's portrayed in the popular
literature as fascinating and interesting, a numerical equivalent of
a rock star. But why? Is it intrinsic?
With best regards,
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