Representation numbers for sums of squares

Prove that

where $r(n)$ is the number of distinct ways of writing $n$ as a sum of two integer squares.

Draw the figure for some values of the variables, count the solutions in integers of the equation

in all cases. Interesting cases are

Let $n$ be a positive integer,

where $2, p_1,\ldots ,p_\mu ,q_1,\ldots ,q_\nu$ are distinct primes, $p_1\equiv \cdots \equiv p_\mu \equiv 1$, $q_1\equiv \cdots \equiv q_\nu \equiv 3 \pmod4$. Set

if $l_1,l_2,\ldots ,l_\nu$ are all even, and $\delta (n)=0$ otherwise. Show that $\delta (n)$ is, in fact, the difference between the numbers of two different kinds of divisors of $n$:

where the summation is extended over the odd divisors $t'$ of $n$.

($4\delta (n)$ is the number of lattice points on the circle $x^2+y^2=n$. This fact was discovered by Gauss.)