## Riemann/PNT

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• 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
Message 1 of 3 , Apr 1, 2006
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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.

Cheers,
Tom
• ... As you know, Gauss was the first to establish that the number of primes
Message 2 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 3 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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