Von Mangoldt function

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

Hands-on definition

The von Mangoldt function, denoted $\mathrm{\Lambda }$, is an arithmetic function defined as follows:

• $\mathrm{\Lambda }(1)=0$.
• $\mathrm{\Lambda }(p^k)=\log p$, for $p$ a prime and $k$ a positive integer.
• $\mathrm{\Lambda }(n)=0$ if $n$ has more than one prime divisor.

Definition in terms of Dirichlet product

The von Mangoldt function is the unique function $\mathrm{\Lambda }$ such that:

$\mathrm{\Lambda }*U=\log$

where $\log$ is the logarithm, $U$ is the all ones function, and $*$ denotes Dirichlet product.

By the Mobius inversion formula, this is equivalent to defining:

$\mathrm{\Lambda }:=\log *\mu$,

where $\mu$ is the Mobius function.