Consider $f(n)$ to be the number of representations $n=x^2 + y^2$ for integral $x$ and $y$. Find the average number of such representations for a natural number, i.e.

\[\lim_{n \to \infty}\frac {f(1) + f(2) + ... + f(n)} {n}\]

## average number of square sum representation

### average number of square sum representation

"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

### Re: average number of square sum representation

It's easy to find that, if $P(n) = f(1) + f(2) + \cdots + f(n)$, then, P(n) is the number of integral solution $(x,y)$ such that $x^2 + y^2 \leq n$ that is, $P(n)$ is the number of lattice points in the circle with radius $\sqrt(n)$ which has it's center in $(0,0)$. This is actually Gauss circle problem, And $\pi n < P(n) \leq \pi n + 2\pi \sqrt(2n)$, which was proved by Gauss. So, The limit is $\pi$

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### Re: average number of square sum representation

Good proof.

I actually collected the problem from a seminal article submitted by a MIT student. That article had some really nice results

I actually collected the problem from a seminal article submitted by a MIT student. That article had some really nice results

"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor