# Mobius 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

The Mobius function is an integer-valued function defined on the natural numbers as follows. The Mobius function at , denoted , is defined as:

• .
•  if  are pairwise distinct primes.
•  if  is divisible by the square of a prime.

### Definition in terms of Dirichlet product

The Mobius function  is defined as the inverse, with respect to the Dirichlet product, of the all ones function , which is defined as the function sending every natural number to . In other words:

.

Here,  is the identity element for the Dirichlet product, and is the function that is  at  and  elsewhere.

## Dirichlet series

The Dirichlet series of the Mobius function is given by:

.

This is equal to the reciprocal of the Riemann zeta-function, because the Mobius function is the inverse of the all ones function with respect to the Dirichlet product. In particular, the function is absolutely convergent for , and it has an analytic continuation to , with its poles being the zeros of the Riemann zeta-function and its zeros being the poles of the Riemann zeta-function.

## Facts

### Mobius inversion formula

Further information: Mobius inversion formula

In terms of Dirichlet products, the Mobius inversion formula states that:

.

The group-theoretic proof of this involves taking the Dirichlet product of both sides with .

In more explicit terms, it states that:

.