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Topic: Goldbach conjecture
Replies: 43   Last Post: Sep 26, 2000 8:55 AM

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Robert Hill

Posts: 529
Registered: 12/8/04
Re: Goldbach conjecture
Posted: Nov 5, 1997 12:23 PM
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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 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 No, but it was already out of date in 1966 :the bound given is Skewes'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)






Date Subject Author
10/29/97
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10/29/97
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feldmann@bsi.fr
11/4/97
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11/13/97
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11/6/97
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11/5/97
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Robert Hill
10/30/97
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11/2/97
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Orjan Johansen
11/2/97
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John Rickard
8/25/98
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Elijah Bishop
11/2/97
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10/30/97
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11/2/97
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11/6/97
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4/28/99
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Papus

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