The normalized two-square count is multiplicative with the expected prime-power values

The ordered-sign representations of $1$ are $(\pm 1,0)$ and $(0,\pm 1)$, so $r_2(1)=4$ and therefore $f(1)=1$.

Write a positive integer $N$ as $N=2^{\alpha }{\prod }_{i=1}^r{p}_i^{e_i}{\prod }_{j=1}^s{q}_j^{b_j},$ where the $p_i\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, the $q_j\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, and all listed primes are distinct. If some $b_j$ is odd and $N=x^2+y^2$, write $N=q_j^{2k+1}M$ with $q_j\nmid M$. Repeatedly applying A prime congruent to $3$ modulo $4$ divides both coordinates of a divisible two-square sum shows that $x=q_j^k{x}_0$ and $y=q_j^k{y}_0$, so $q_{j}M=x_0^2+y_0^2$. One more application of the same lemma forces $q_j\mid x_0$ and $q_j\mid y_0$, hence $q_j^2\mid q_{j}M$, contradiction. Therefore $r_2(N)=0$ whenever some $b_j$ is odd. If every $b_j$ is even, the cited Hackman theorem gives $\frac{r_2(N)}{4}={\prod }_{i=1}^r(e_i+1)\mathrm{.}$ Applying this to $N=2^a$, $N=p^a$, and $N=q^a$ yields $\frac{r_2(2^a)}{4}=1,\frac{r_2(p^a)}{4}=a+1,\frac{r_2(q^a)}{4}=\{\begin{array}{ll}1,& a\text{ even},\\ 0,& a\text{ odd}\mathrm{.}\end{array}$

Let $f(n):=r_2(n)\mathrm{/}4$. If $\gcd (m,n)=1$, the prime supports of $m$ and $n$ are disjoint. If step 2.1 gives $f(m)=0$ or $f(n)=0$, then some prime $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ has odd exponent in one factor, hence still odd exponent in $mn$, so step 2.1 also gives $f(mn)=0=f(m)f(n)$. Otherwise every prime $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ occurs to even exponent in both $m$ and $n$, so also in $mn$, and the primes $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ occurring in $mn$ are exactly those occurring in one factor or the other, with the same exponents. Step 2.1 therefore factors the nonzero case as $f(mn)={\prod }_{p^e\parallel m,p\equiv 1(4)}(e+1){\prod }_{p^{e^{\mathrm{\prime }}}\parallel n,p\equiv 1(4)}(e^{\mathrm{\prime }}+1)=f(m)f(n)\mathrm{.}$ Together with $f(1)=1$ from step 1.1, this is exactly the multiplicativity condition of Multiplicative arithmetic functions.

Steps 1.1, 2.1, and 3.1 prove the stated prime-power values and multiplicativity.