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badreferences
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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!
 2 years ago
 2 years ago
badreferences Group Title
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!
 2 years ago
 2 years ago

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badreferences Group TitleBest ResponseYou've already chosen the best response.0
Furthermore, using Mathematica to algorithmically evaluate \(x\) by using arbitrarily large values of \(n\) does not converge to \(1\)!
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.4
* i love to know what is the answer
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.4
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
 2 years ago

badreferences Group TitleBest ResponseYou've already chosen the best response.0
It's only necessary that someone links to a proof. I know I've seen this before, somewhere.
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.1
This is called the prime number theorem.
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.1
http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf
 2 years ago

badreferences Group TitleBest ResponseYou've already chosen the best response.0
Oh, wait, I found it in mathworld. Okay, thanks!
 2 years ago

badreferences Group TitleBest ResponseYou've already chosen the best response.0
That's right. This is Legendre's constant. That's where I saw it before! http://en.wikipedia.org/wiki/Legendre%27s_constant
 2 years ago

badreferences Group TitleBest ResponseYou've already chosen the best response.0
Thanks for hte link, @eliassaab
 2 years ago
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