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Re: Number of prime numbers between the values of (m) (m1)

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  • djbroadhurst
    ... It is interesting to see how Sergey fooled himself into making his false conjecture. His mistake was to believe that we may rely on the sieve of
    Message 1 of 5 , Jul 7, 2010
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > m*prod(k=1,pi(sqrt(m)),1 - 1/prime(k)) + n - 1 = pi(m) ... [1]
      > It is rather easy to prove that the number of solutions to [1]
      > is finite, in direct contradiction to a conjecture by Sergey.

      It is interesting to see how Sergey fooled himself into
      making his false conjecture. His mistake was to believe
      that we may rely on the sieve of Eratosthenes to give
      the "right" constant in the prime number theorem. In fact,
      we should not: there is a mismatch by the celebrated factor
      2*exp(-Euler) > 1 that so perplexed Pafnuty Lvovich Chebyshev
      and Franz Carl Joseph Mertens.

      Elsewhere in this list, one may find an investigation of this issue.
      In particular, my remarks in
      http://tech.groups.yahoo.com/group/primenumbers/message/20936?var=0
      prompted Mike Oakes' impressive statistics in
      http://tech.groups.yahoo.com/group/primenumbers/message/20940?var=0

      David
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