Arithmetic consequences of Jacobi's two-squares theorem (Q1567061)

Let \(r_2(n)\) denote the number of representations of \(n\) as a sum of two squares. Jacobi's formula for \(r_2(n)\) easily implies: \[ \begin{alignedat}{2} r_2(2n)&= r_2(n), &&\\ r_2(pn)&= 2r_2(n)- r_2 \bigl( \tfrac{n}{p} \bigr) &\quad\text{for }&p\in\mathbb{P},\;p\equiv 1\bmod 4,\\ r_2(pn)&= r_2 \bigl( \tfrac{n}{p} \bigr) &\quad\text{for }&p\in\mathbb{P},\;p\equiv 3\bmod 4.\end{alignedat} \] These formulas are proved directly via generating functions. They easily imply Jacobi's two-square theorem.