Mobius function

From Number
Revision as of 21:20, 22 April 2009 by Vipul (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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


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.


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: