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badreferences

Prove that \(x\) is both rational and an integer.\[\lim_{n\to\infty}\left(\ln n-\frac{n}{\pi\left(n\right)}\right)=x\]The function \(\pi(n)\) counts the number of primes less than or equal to \(n\). I've seen this equation somewhere before, but I can't remember where. IIRC, \(x\) evaluates to \(1\). Mysterious!

  • one year ago
  • one year ago

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  1. badreferences
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    Furthermore, using Mathematica to algorithmically evaluate \(x\) by using arbitrarily large values of \(n\) does not converge to \(1\)!

    • one year ago
  2. mukushla
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    * i love to know what is the answer

    • one year ago
  3. mukushla
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    i found something similar \[\lim_{n\to\infty}\frac{\pi(n)}{\frac{n}{\ln n}}=1\] http://en.wikipedia.org/wiki/Prime_number_theorem http://mathworld.wolfram.com/PrimeNumberTheorem.html

    • one year ago
  4. badreferences
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    It's only necessary that someone links to a proof. I know I've seen this before, somewhere.

    • one year ago
  5. eliassaab
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    This is called the prime number theorem.

    • one year ago
  6. eliassaab
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    http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf

    • one year ago
  7. badreferences
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    Oh, wait, I found it in mathworld. Okay, thanks!

    • one year ago
  8. badreferences
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    That's right. This is Legendre's constant. That's where I saw it before! http://en.wikipedia.org/wiki/Legendre%27s_constant

    • one year ago
  9. badreferences
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    Thanks for hte link, @eliassaab

    • one year ago
  10. eliassaab
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    yw

    • one year ago
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