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Re: Number of factors

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  • djbroadhurst
    ... For log(p)
    Message 1 of 7 , Oct 3, 2011
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      --- In primenumbers@yahoogroups.com,
      Bernardo Boncompagni <RedGolpe@...> wrote:

      > > log(log(N)) + B_1 - sum(prime q < p, 1/q) + o(1)
      > Interestingly enough, if p>>1, sum=B_1+log(log(p))
      > and we get log(log(N))-log(log(p)).

      For log(p) << log(N), the average of omega(n/p),
      with n running from p to N and p the least
      prime divisor of n, is (I claim) asymptotic to the
      sum of 1/q, with prime q running from q = p up to
      the greatest prime q <= N/p.

      All I did was to use Mertens at the upper limit.
      For log(2) << log(p) << log(N), one may also use
      Mertens at the lower limit to get log(log(N)/log(p)),
      as Bernardo has observed. However one should not push
      this up to p = O(sqrt(N)), since there we may run into
      a problem that worried Chebyshev:

      David (per proxy Pafnuty Lvovich)
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