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 Möbius Inversion Formula we have that $\Lambda = \mu * \ln$. $\blacksquare$