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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).
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Contents 1 Definition 1.1 Definition in terms of Dirichlet product 2 Dirichlet series 3 Facts 3.1 Mobius inversion formula

Definition

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

• μ(1) = 1.
• if p_i are pairwise distinct primes.
• μ(n) = 0 if n is divisible by the square of a prime.

Definition in terms of Dirichlet product

The Mobius function μ is defined as the inverse, with respect to the Dirichlet product, of the all ones function U, which is defined as the function sending every natural number to 1. In other words:

μ * U = U * μ = I.

Here, I is the identity element for the Dirichlet product, and is the function that is 1 at 1 and 0 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 μ.

In more explicit terms, it states that:

.