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• ... As you know, Gauss was the first to establish that the number of primes
Message 1 of 3 , Apr 1, 2006
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In an email dated Sun, 2 4 2006 1:10:54 am GMT, "gulland68" <tmgulland@...> writes:

>Can anyone explain to me in layman's terms what the proof of the
>Riemann hypothesis would say about the distribution of primes that the
>Prime Number Theorem does not? I'm working through the John Derbyshire
>book but I fear it won't quite answer that question, save in rather
>too cryptic/tacit terms.

As you know, Gauss was the first to establish that the number of primes < x, pi(x), is approximately equal to the logarithmic integral of x, Li(x), for large x.

The Prime Number Theorem gives a bound on the difference, which is roughly O(x/ln(x)).

If the Riemann Hypothesis were to be proved, then that bound could be considerably tightened, to O(sqrt(x)*ln(x)).

All this is nicely explained at, for example:

Hope this helps.

-Mike Oakes
• If you could prove that the convergence upon Li(x) (or even x/log(x)) becomes increasingly such (i.e., the more convergent it becomes, the more convergent it
Message 2 of 3 , Apr 3, 2006
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If you could prove that the convergence upon Li(x) (or even x/log(x))
becomes increasingly such (i.e., the more convergent it becomes, the
more convergent it becomes)- then surely you could derive the proof of
the Weak Merten's Conjecture from it, on the basis of
the times-squared law...?

Cheers,

Tom
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