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!

Question

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!

anonymous · Answer

Furthermore, using Mathematica to algorithmically evaluate $x$ by using arbitrarily large values of $n$ does not converge to $1$!

anonymous · Answer

* i love to know what is the answer

anonymous · Answer

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

anonymous · Answer

It's only necessary that someone links to a proof. I know I've seen this before, somewhere.

anonymous · Answer

This is called the prime number theorem.

anonymous · Answer

http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf

anonymous · Answer

Oh, wait, I found it in mathworld. Okay, thanks!

anonymous · Answer

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

anonymous · Answer

Thanks for hte link, @eliassaab

anonymous · Answer

yw