## anonymous 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!

1. anonymous

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

2. anonymous

* i love to know what is the answer

3. 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

4. anonymous

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

5. anonymous

This is called the prime number theorem.

6. anonymous
7. anonymous

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

8. anonymous

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

9. anonymous