Möbius function

The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted μ(x).

This article is about the number-theoretic Möbius function. For the combinatorial Möbius function, see incidence algebra. For the rational functions defined on the complex numbers, see Möbius transformation.
Möbius function
Named afterAugust Ferdinand Möbius
Publication year1832
Author of publicationAugust Ferdinand Möbius
No. of known termsinfinite
First terms1, −1, −1, 0, −1, 1, −1, 0, 0, 1
OEIS index
  • A008683
  • Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.
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