# 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).
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## 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.