The von-Mangoldt Function
 Definition: The von-Mangoldt Function is the function $\Lambda : \mathbb{N} \to \mathbb{R}$ defined for all $n \in \mathbb{N}$ by $\displaystyle{\Lambda (n) = \left\{\begin{matrix} \ln p & \mathrm{if} \: n = p^k\\ 0 & \mathrm{otherwise} \end{matrix}\right.}$.
Sometimes the notation "$\log p$" is used in place for "$\ln p$", in which case it is understood that "$\log p$" has base $e$.
Some of the first few values of $\Lambda (n)$ are given in the table below.
 $n$ $\Lambda (n)$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $0$ $\ln 2$ $\ln 3$ $\ln 2$ $\ln 5$ $0$ $|ln 7$ $\ln 2$ $\ln 3$ $0$
 Proposition 1: $\displaystyle{\Lambda = \mu * \ln}$.
• Proof: It can easily be shown that $\ln = \Lambda *1$. Since $\ln(n) = \sum_{d|n} \Lambda (d)$, by The Möbius Inversion Formula we have that $\Lambda = \mu * \ln$. $\blacksquare$