Multiplicative Property of $A(n)$

$$A(n)=\sum _{d\mid n}\chi (d)=\sum _{d\mid n}\chi (d)u\left(\frac{n}{d}\right)=\chi \ast u.$$

Since $\chi$ is multiplicative and $\mu$ is multiplicative, their Dirichlet product is also multiplicative. See Theorem 2.14 for proof.