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A Logarithmic Property of Number of Primes between n^p and n^(p+c)

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  • Anton
    Hello I unfortunately have no good books on the prime number theorem and my studies in prime numbers have led me to this conjecture: The difference between the
    Message 1 of 1 , Sep 7, 2005
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      Hello

      I unfortunately have no good books on the prime number theorem and my
      studies in prime numbers have led me to this conjecture:

      The difference between the logarithm of the prime count below a
      number n raised to the power of p and the logarithm of the prime
      count below the same number n raised to p+c divided by the the
      product of c and the logarithm of n the result approaches 1-1/pLog(n)
      or is asymptotic 1-1/pLog(n)


      Log[pi(n^(p+c))] - Log[pi(n^p)]
      ------------------------------- ~ 1 - 1/(p Log[n])
      c Log[n]

      try it yourself

      is this new?
      can you explain or proove it?

      some examples for above
      the left hand side
      n=10, p=22, c=1 evaluates to 0.98029847
      n=10, p=20, c=3 evaluates to 0.97932991
      n=2, p=35, c=2 evaluates to 0.95866205

      and the right hand side
      n=10 p=22, c=1 evaluates to 0.980259342
      n=10 p=20, c=3 evaluates to 0.978285276
      n=2, p=35,c=2 evaluates to 0.958780142
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