
Re: Goldbach conjecture
Posted:
Nov 5, 1997 12:23 PM


In article <878561083.20928@dejanews.com>, feldmann@bsi.fr writes: > 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 uptodate? > > No, but it was already out of date in 1966 :the bound given is Skewes' 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
According to the zeta function poster distributed by Wolfram to advertise Mathematica, the first change of sign is now known to be < 10^400.
I have a vague idea that more specific information and references can be found in the survey paper by Odlyzko on computations in analytic number theory, in a fairly recent volume in the Proc. Symposia App. Math. series, that marked a major anniversary of Mathematics of Computation. The volume is mainly about numerical computation generally but contains a substantial section of several papers on number theoretic computation.
 Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, the heart is true."  John Dowland, Fine Knacks for Ladies (1600)

