Re: [PrimeNumbers] Riemann/PNT
- 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 theAs 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.
>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.
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.
- 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...?