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Re: Number of primes in the interval [a ^ 2, b ^ 2]. b-a = 1

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  • djbroadhurst
    ... The number Q of such primes is asymptotic to a/log(a). Sergey over-estimates Q by the well-known percentage 2*exp(-Euler) - 1 =~ 12.29% from bad reasoning
    Message 1 of 2 , Dec 15, 2010
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      --- In primenumbers@yahoogroups.com,
      <chitatel2000@...> wrote:

      > Number of primes in the interval [a ^ 2, b ^ 2]. b-a = 1

      The number Q of such primes is asymptotic to a/log(a).

      Sergey over-estimates Q by the well-known percentage
      2*exp(-Euler) - 1 =~ 12.29%
      from bad reasoning about the sieve of Eratosthenes.

      Example: Set a = 10^6.
      There are 37,607,912,018 primes less than 10^12.
      There are 37,607,984,431 primes less than 10^12 + 2*10^6 +1.
      Here, Q = 72413 is indeed close to 10^6/log(10^6) =~ 72382.

      Now look at Sergey's false estimate:
      (2*10^6 + 1)*prod(p=2,10^6,1-if(isprime(p),1./p)) =~ 81276.

      This leaves him with the egregious error
      E = 81276 - 72413 = 8863, i.e. 12.24% of Q, which is indeed
      close to the Mertens estimate, 12.29% of Q, provided above.

      David
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