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While reading this post, I stumbled across these definitions (Wiki_german)

$$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$

and

$$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$

The last one seems interesting, since $ \lim_{n \to \infty} (\sqrt[n]{n})=1$, proven here.

How to prove these?

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    In the second one, you can use that $\pi(n)\sim n/\log(n)$ asymptotically.2012-01-05
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    Regarding your comment, note that $\pi(n)\to\infty$, so in order for $\sqrt[n]{n}^{\pi(n)}$ to have a finite limit it is necessary for $\sqrt[n]{n}$ to converge to $1$.2012-01-05
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    @Raskolnikov: So infact one doesn't really need $\pi(n)$ to prove it?2012-01-05
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    Let's just say that it is an awkward way to rephrase a property of the [prime-counting function](http://en.wikipedia.org/wiki/Prime-counting_function). Same is probably true for the primoral $n\#$.2012-01-05
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    Definitions? Of what are those two formulas "definitions"? Anyway, the product of the primes up to $n$ is asymptotic to $e^n$ - I believe this is at the same level of difficulty as the Prime Number Theorem - that should get you the first equation.2012-03-26

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