In article <email@example.com>, firstname.lastname@example.org writes: > In article <rbloomEJ1w13.CyD@netcom.com>, > email@example.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 No, but it was already out of date in 1966 :the bound given is Skewes' 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
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)