Mobius function
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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
Contents
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.
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:
.
