Given $x\in \mathbb{C}$ let's define the function $$\cases { \omega_x(n)=1, \quad n=1 \\ \omega_x(n)=x^r, \quad n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}}$$
a) Prove that $\omega_x(n) \colon \mathbb{N} \longrightarrow \mathbb{C} $ is multiplicative
b) Develop a closed-form for $F(n) = \sum_{d|n} \omega_x(d) $
I think I found a closed-form in order to solve the part b) but I am not sure if it is correct.
I know that, given $\omega_x(n)$ multiplicative, $F(n)$ is multiplicative too, so I split it into the $r$ parts of the factorization of $n$ $$F(p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r})= F(p_1^{a_1})F(p_2^{a_2}) \cdots F(p_r^{a_r})$$ $$F(p_1^{a_1})=\omega_x(1)+\omega_x(p_1)+\omega_x(p_1^2)+\cdots +\omega_x(p_1^{a_1})= 1+x+x+\cdots+x=1+a_1x$$
So
$$F(p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r})= \prod_{i=1}^r(1+a_ix)$$
Is it correct? Thank you for your help.