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:

http://mathworld.wolfram.com/PrimeNumberTheorem.html
Hope this helps.

-Mike Oakes