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Re: [PrimeNumbers] Riemann/PNT

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  • mikeoakes2@aol.com
    ... 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:
      http://mathworld.wolfram.com/PrimeNumberTheorem.html

      Hope this helps.

      -Mike Oakes
    • gulland68
      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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