# Relation for multiplicative functions

I have some problems understanding relations between convolutions and Euler product. I want to express as a convolution or an additive formulation for:

$$N\varphi(N) \prod_{p | N} (1+p^{-1})$$

Expanding in product the Euler function $\varphi$, we get:

$$N^2 \prod_{p | N} (1-p^{-2})$$

Could I just expand it roughly and get that it is equal to:

$$\mu \star \mathrm{id}^2 (N) \quad ?$$

Is there a general method to "see" the relations between those types of products, sums and convolutions?