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