From: feldmann@bsi.fr
Subject: Re: Goldbach conjecture
Date: Mon, 03 Nov 1997 06:51:18 -0600
Newsgroups: sci.math

In article <rbloomEJ1w13.CyD@netcom.com>,
  rbloom@netcom.com (Ron Bloom) wrote:
>
>                 /^\
>                 |
>   Let Li(x) =   |   du/ln(u)   "The logarithmic integral"
>                 |
>               \_/
>                0
>
>   Let Pi(x) = "The number of primes, p <= x "
>
>   It is known that  Li(x)/Pi(x) ---> 1  as x --> oo.
>
>   It is known that Li(x) > Pi(x) infinitely often
> and                Li(x) < Pi(x) infinitely often
>
> It is _not_ known how large x must be before we encounter
> the first place where Li(x) <= Pi(x).   An upper bound on
> such an x is exp(exp(exp(79)))
>
> Source:  Excursions in Number Theory, Ogilvy & Anderson, 1966
>
>  Question:  is this info still up-to-date?

No, but it  was already out of date in 1966 :the bound given is Skewes'
number (1955, thought to be the highest number appearing in a "real"
theorem for a long time)  The bound was lowered to 10^(10529) in 1964,
then again, by Lehman, to 10^(1130) in 1966. In 1985, William and Fern
Ellisson (Prime Numbers) had no better lower bound, and I dont think
substantial progress have been made... More interesting results (like
number of sign changes in Li(x)-Pi(x)) are given in the same book. Hope
the information will be useful