0
$\begingroup$

The question is self-explainatory.

For example are there any known identities involving Euler Totient function and $\pi$ ?

$\endgroup$
  • $\begingroup$ From wikipedia: $$ \sum_{k= 1}^n\varphi(k) = \frac{3n^2}{\pi^2} + O\left(n(\log n)^{2/3}(\log\log n)^{4/3}\right) $$ $\endgroup$ – Stahl Dec 28 '13 at 17:12
2
$\begingroup$

There are several identities of this kind. Here are just some examples: $$ \sum_{n=1}^{\infty}\frac{\mu(n)}{n^2}=\frac{6}{\pi^2}, \; \sum_{n=1}^{\infty}\frac{\mu(n)}{n^4}=\frac{90}{\pi^4},\; \ldots , $$ where $\mu(n)$ denotes the Moebius $\mu$-function; $$ \lim_{n\to \infty}\frac{\sigma(1)+\cdots \sigma(n)}{n^2}=\frac{\pi^2}{12}, $$ where $\sigma(n)$ denotes the sum of divisor function; $$ \lim_{n\to \infty}\frac{\phi(1)+\cdots \phi(n)}{n^2}=\frac{3}{\pi^2}, $$ where $\phi(n)$ denotes Euler's $\phi$-function.

$\endgroup$
1
$\begingroup$

Sterling formula for finding limit is another example that utilizes pi. And to make your question more " appealing " you might want to add another ingredient like e and states it like this: " list all possible identities and approximations that involve either pi or e or both constants "

$\endgroup$
1
$\begingroup$

If $f(n)$ ($n=1,2,3,\ldots$) is the number of ways that $n$ can be written in the form $n=a^2 + b^2$, with $a,b\in\mathbb{Z}$, then there is a very nice fact about the average value of this function:

$$\lim_{n\to\infty} \frac{f(1)+f(2)+\cdots + f(n)}{n} = \pi$$

Mysterious, or obvious? I'll leave that up to you.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.