The Sum-of-Squares Function and Jacobi's Two-Square Theorem

This entry defines the sum-of-squares function $r_k(n)$, which counts the number of ways to write a natural number $n$ as a sum of $k$ squares of integers. Signs and permutations of these integers are taken into account, such that e.g. $1^2+2^2$, $2^2+1^2$, and $(-1{)}^2+2^2$ are all different decompositions of $5$.

Using this, I then formalise the main result: Jacobi's two-square theorem, which states that for $n>0$ we have $$r_2(n)=4(d_1(3)-d_3(n)),$$ where $d_i(n)$ denotes the number of divisors $m$ of $n$ such that $m=i(\text{mod}4)$.

Corollaries include the identities $r_2(2n)=r_2(n)$ and $r_2(p^{2}n)=r_2(n)$ if $p=3(\text{mod}4)$ and the well-known theorem that $r_2(n)=0$ iff $n$ has a prime factor $p$ of odd multiplicity with $p=3(\text{mod}4)$.