average number of square sum representation

For college and university level advanced Mathematics
User avatar
Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

average number of square sum representation

Unread post by Avik Roy » Tue Mar 29, 2011 6:53 pm

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.
\[\lim_{n \to \infty}\frac {f(1) + f(2) + ... + f(n)} {n}\]
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

Corei13
Posts: 153
Joined: Tue Dec 07, 2010 9:10 pm
Location: Chittagong

Re: average number of square sum representation

Unread post by Corei13 » Wed Sep 07, 2011 3:29 am

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 $\pi$
ধনঞ্জয় বিশ্বাস

User avatar
Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

Re: average number of square sum representation

Unread post by Avik Roy » Wed Sep 07, 2011 3:29 pm

Good proof.
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

Post Reply