[FOM] a lower bound? (another ps to my reply to Bob Solovay.)

Gabriel Stolzenberg gstolzen at math.bu.edu
Thu Apr 20 12:31:58 EDT 2006

   I wanted to write about this in my previous message but I found
it too confusing.

                         A lower bound?

   I wonder why I haven't seen anything about a lower bound for a
sign change of pi - li.  I would have thought that this would be of
greater interest to folks like Littlewood than an upper bound.  As I
understand it, Gauss conjectured that a certain relationship that he
observed for low values, pi(x) < li(x), persists forever.  Just as
he conjectured that the estimate that he noticed for the density of
primes for low values persists forever.

   But if so, wouldn't it have been more interesting to Littlewood et
al to find a value larger than those that Gauss observed up to which
the relationship pesists?  Even if it turned out to be far from the
first sign change, wouldn't it make Littlewood's refutation of Gauss's
conjecture seem that much more impressive?

      Does the prime number theorem confirm what Gauss observed?

   I don't know of any proof of the prime number theorem that shows
that the error estimate (as a function of n) that Gauss observed
for relatively small values presists forever.  They show only that
the estimate holds for sufficiently large n.  I think it was Rosser
who dealt with the middle range, getting an estimate that holds from
59 up to some sufficiently large number.

   If I have this right (and I may not), I wonder why proofs of the
prime number theorem are often said to confirm Gauss's conjecture,
while Rosser's work, which, as I see it, is the last step in doing
so, receives very little attention?

    With best regards,

   Gabriel Stolzenberg

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