# Von Mangoldt function

From Number

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 , is an arithmetic function defined as follows:

- .
- , for a prime and a positive integer.
- if has more than one prime divisor.

### Definition in terms of Dirichlet product

The von Mangoldt function is the unique function such that:

where is the logarithm, is the all ones function, and denotes Dirichlet product.

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

,

where is the Mobius function.