[FOM] reply to bob solovay

Gabriel Stolzenberg gstolzen at math.bu.edu
Mon Apr 17 23:28:44 EDT 2006

On Sun, 16 Apr 2006, Robert M. Solovay wrote:

> On Sun, 16 Apr 2006, Gabriel Stolzenberg wrote:
> > (Skewes began sometime between 1912 and 1933 and finished in 1955,
> > three years after Kreisel explained how to read a bound out of
> > Littlewood's proof.)
> Skewes bound was quite explicit [though large]. Would K. get such an
> explicit bound. How would the size of K.'s bound compare to that of
> Skewes?
>  	--Bob Solovay

   Good questions.  Kreisel didn't actually read such a bound out of
Littlewood's proof.  He just made some intelligent remarks about how
to do it.  However, in 1973, my student, Bill Hruschka, did---as part
of his doctoral thesis, "A constructive critique of "The Distribution
of Prime Numbers" by A. E. Ingham, with particular reference to
Littlewood's theorem about p(x) - li(x)."  But Bill made no attempt to
get an explicit bound to compare with the bounds of Skewes and Lehman.
Our interests were elsewhere.

   Would it be feasible to go further and get an explicit bound from
Bill's thesis?  I have no idea.  (It's been more than 30 years since
I've thought about this stuff.)  But because the constructive proof
yields a very complicated definition of a real number, in order to get
a finite decimal above it, we will have to make many approximations.
And, for all I know, the particular ways we make them---there will be
choices---could make a significant difference in the size of the finite
decimal that we get (and even in the feasibility of the computation).

   Thanks again for the good questions.  I'm sorry I can't be of more

    With best regards,

   Gabriel Stolzenberg

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