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

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:

.