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badreferences

  • 3 years ago

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!

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

  2. mukushla
    • 3 years ago
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    * i love to know what is the answer

  3. mukushla
    • 3 years ago
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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

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

  5. eliassaab
    • 3 years ago
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    This is called the prime number theorem.

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

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

  8. badreferences
    • 3 years ago
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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

  9. badreferences
    • 3 years ago
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    Thanks for hte link, @eliassaab

  10. eliassaab
    • 3 years ago
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    yw

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