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

The question is self-explainatory.

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

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

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