The von-Mangoldt Function

# 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.

$n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|

$\Lambda (n)$ | $0$ | $\ln 2$ | $\ln 3$ | $\ln 2$ | $\ln 5$ | $0$ | $|ln 7$ | $\ln 2$ | $\ln 3$ | $0$ |

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 MÃ¶bius Inversion Formula we have that $\Lambda = \mu * \ln$. $\blacksquare$