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Topic: Spencer-Brown's purported proof of Riemann Hypothesis
Replies: 8   Last Post: Aug 7, 2006 12:17 PM

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john baez

Posts: 460
Registered: 12/6/04
Re: Spencer-Brown's purported proof of Riemann Hypothesis
Posted: Aug 6, 2006 6:05 AM
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In article <je2dnSC6K7IEWVbZnZ2dnUVZ_qWdnZ2d@comcast.com>,
Tim Peters <tim.one@comcast.net> wrote:

>He claims that these first few pages prove:
>
> |pi(n) - li(n)| < sqrt(li(n)) [3]
> for all n > 1


>Anyway, /if/ [3] holds RH follows, because:
>
>a. sqrt(li(n)) < sqrt(n) for n > 1 (which he notes)
>
>b. so |pi(n) - li(n)| < sqrt(n) for n > 1 would follow from [3]
>
>c. RH is equivalent to the statement (which he alludes to, but doesn't
> give) that there's some constant C for which:
>
> |pi(n) - li(n)| < C sqrt(n) log(n)
>
> holds for all sufficiently large n; or, IOW,
>
> pi(n) = li(n) + O(sqrt(n) log(n))
>
>So [3], if true, is in fact a substantially tighter bound than required to
>prove RH.


At first I thought it might be too tight to be true - but now
I guess such a bound has not yet been ruled out.

In Bombieri's article on the Riemann Hypothesis:

http://www.claymath.org/millennium/Riemann_Hypothesis/Official_Problem_Description.pdf

he writes:

"The validity of the Riemann hypothesis is equivalent to saying
that the deviation of the number of primes from the mean Li(x) is

pi(x) = Li(x) + O(sqrt(x) log x);

the error term cannot be improved by much, since it is known to
oscillate in both directions to order at least Li(sqrt(x)) log log log x
(Littlewood)."

But, unless I'm making a mistake, Li(sqrt(x)) log log log x
is eventually much smaller than Spencer-Brown's claimed bound sqrt(Li(x)).

Of course this has nothing to do with whether his "proof" makes sense.




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