Mangoldt

The Mangoldt function is defined by the formula $$\Lambda(n) = \left\{ \begin{array}{ll} \log p &; n=p^k \mathrm{\hspace{2pt}for\hspace{2pt}some\hspace{2pt}prime\hspace{2pt}}p\mathrm{\hspace{2pt}and\hspace{2pt}integer\hspace{2pt}}k\geq 1, \\ 0 &; \mathrm{otherwise}. \end{array} \right.$$

Properties

Theorem: The following formula holds: $$\log(n) = \displaystyle\sum_{d | n} \Lambda(d),$$ where $\log$ denotes the natural logarithm and the notation $d | n$ denotes that $d$ is a divisor of $n$.

Proof: █

Videos

Number Theory 31: Liouville and mangoldt functions
Number theory: Arithmetic functions #1