how to prove $f$ is an arithmetic function with this property $\sum_{d\mid n} f(d)=n^2$

Question

how to prove $f$ is an arithmetic function with this property $$\sum_{d\mid n} f(d)=n^2$$

Arithmetic function

Answer 1

I think by saying arithmetic function,OP means that $f(1)=1$ and for all $a,b,$ $$(a,b)=1 \Rightarrow f(a)f(b)=f(ab).\tag1$$ From Möbius inversion formula as Frank Science commented, $$f(n)=n^2\sum_{d|n}{\dfrac{u(d)}{d^2}}=n^2\prod_{p|n}{(1-\frac{1}{p^2})},\tag2$$ for every $n\geq1.$

Now it's obvious that $f(1)=1$,and $(1)$ is hold.Hence $f(n)$ is an arithmetic function.