[FOM] part III of my comments on Harvey's comments on "Harvey's effective number theorists."

Gabriel Stolzenberg gstolzen at math.bu.edu
Sun Apr 16 16:25:06 EDT 2006

   This concludes my comments on Harvey's comments on "Harvey's
effective number theorists."

   Harvey quotes me as saying,

> > If, as Harvey suggests and his evidence seems to support, there is
> > an "intrinsic" interest in "effective" bounds and algorithms, then
> > I'm disappointed.

   He then remarks,

> I have been curious to know why you are disappointed. I am elated.

   Later, Harvey mentions that his unnamed number theorist considers
the problem of getting an "effective" algorithm in a certain case
"obviously" fundamental and is probably unwilling to say any more
about it.  Well, my simple answer as to why I'm disappointed is much
the same.  From where I stand, it's obvious.

> > I wonder what Harvey would have made of the case of a sign change
> > for pi - li, with Littlewood's classical proof and Skewes' number
> > as the bound.  For a time, it was the most famous case of this kind.

   Harvey then quotes from a website about primes.

> "Riemann and Gauss believed that Li(x) > p(x) for every x > 3, and all
> the computed values of Li(x) - p(x) today satisfy this inequality.
> However it has been proved by Littlewood in 1914 that Li(x) - p(x)
> changes its sign infinitely  often. In 1933, S. Skewes established
> that the first change of sign occurs before the impressive number 10^k
> with k = 10^10^34 ."

   1933?  Try 1955.  Harvey then discusses several recent seemingly
impressive improvements in the bound, concluding,

> The Li(x) - p(x) project could afford the first known value x for
> which p(x) > Li(x) ! Finding the first change of sign is even more
> difficult !"

   Perhaps somebody can tell me what "finding" means here.

   Harvey continues,

> This is obviously something to be proud of. It starts off with a "not
> desirable bound", which has gradually over the years, become somewhat
> more desirable. There is the big open problem of seeing if the bound
> can be made much better.

   Obviously.  If I were to ask why number theorists are interested in
approximating an x for which p(x) > li(x) or in getting ever better
approximations to the first sign change (e.g., first a "not desirable"
one to within 2, then a "more desirable" one to within 1.5, then an
"even more desirable" one to within 1.25,...), Harvey presumably would
say that it is "obvious."  Which, just as with my "explanation" of my
disappointment, is (even if true) a conversation killer.

> >  I also wonder what Harvey would have made of a brief exchange I
> > once had with Harold Stark about a first cousin to pi - li that he
> > had just proved.  In the exchange, he played the role of Littlewood
> > and the question was whether we also needed a Skewes.

> I can't make anything of this from what you wrote.

   Actually, you should be able to.  You should either say that the
answer is yes---this fits with what you say above about pi - li and
Skewes.  Or, if you take into account that a few years before Skewes
finished, Kreisel explained that it's routine to read bounds out of
Littlewood's proof, you should say that not only did Stark not need a
Skewes, neither did Littlewood.


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