Sum of squares of sum of squares function $r_2(n)$ Let $r_2(n)$ denote the number of representations of $n$ as a sum of two squares.
What is known about the sum of squares of this function,

$\sum_{i=1}^n r_2(i)^2$

In particular is anything known about the asymptotics as $n \rightarrow \infty$?
 A: Asymptotics are known for all integer moments of $r_2$: for any $m \ge 1$, there is an explicit constant $a_m$ such that $\sum_{n \le x} r_2(n)^m \sim a_m\cdot x\,(\log x)^{2^{m-1}-1}$.  ($m=0$ also holds if we interpret $0^m$ as $0$.)
This result, including the precise constant (and a uniform generalization to other positive binary quadratic forms), can be found in V. Blomer and A. Granville, “Estimates for representation numbers of quadratic forms”, Duke Math. J. 135 (2006).  Here's the PDF.
A: If you want to know
exact formulae for 
these types of sums,
I highly recommend
"Number Theory in the Spirit of Ramanujan"
by Bruce Berndt.
On page 56, a proof is given of
$r_2(n) = 4 \sum_{d|n, d\ odd} (-1)^{(d-1)/2}$.
You can do a Dirichlet multiplication
to get a double sum for $r_2^2(n)$.
By the way, after reading this book,
I am absolutely astounded
at how great a mathematician
Jacobi was.
He developed the $q$-series methods
that enabled him to
get explicit formulae for
the number of representations on any integer
as the sum of a number of squares
or triangular numbers.
Reading a list of his accomplishments
is, like wow!
He and Ramanujan would have gotten along famously.
