Definition (von Mangoldt function). The von Mangoldt function $\mathrm{\Lambda }:\mathbb{N}\to ℝ$ is

$$\mathrm{\Lambda }(n)=\{\begin{array}{cc}\log p\hfill & \text{if }n=p^k\text{ for some prime }p\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

Then $\log =1\ast \mathrm{\Lambda }$ since

Since $\mu$ is the inverse of $1$, we have

$$\log \ast \mu =1\ast \mathrm{\Lambda }\ast \mu =\mathrm{\Lambda }\ast 1\ast \mu =\mathrm{\Lambda }\ast I=\mathrm{\Lambda }.$$