anonymous
  • anonymous
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!
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
Furthermore, using Mathematica to algorithmically evaluate \(x\) by using arbitrarily large values of \(n\) does not converge to \(1\)!
anonymous
  • anonymous
* i love to know what is the answer
anonymous
  • anonymous
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

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anonymous
  • anonymous
It's only necessary that someone links to a proof. I know I've seen this before, somewhere.
anonymous
  • anonymous
This is called the prime number theorem.
anonymous
  • anonymous
http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf
anonymous
  • anonymous
Oh, wait, I found it in mathworld. Okay, thanks!
anonymous
  • anonymous
That's right. This is Legendre's constant. That's where I saw it before! http://en.wikipedia.org/wiki/Legendre%27s_constant
anonymous
  • anonymous
Thanks for hte link, @eliassaab
anonymous
  • anonymous
yw

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