Are there any identities linking arithmetic functions and $\pi$?

Question

The question is self-explainatory.

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

Answer 1

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.

Answer 2

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 "

Answer 3

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.