The Möbius function ${\displaystyle \mu (n)}$ 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 ${\displaystyle \mu (x)}$.

The Möbius function is defined by

The Möbius function can alternatively be represented as

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta, ${\displaystyle \lambda (n)}$ is the Liouville function, ${\displaystyle \omega (n)}$ is the number of distinct prime divisors of ${\displaystyle n}$, and ${\displaystyle \Omega (n)}$ is the number of prime factors of ${\displaystyle n}$, counted with multiplicity.

Another characterization by Carl Friedrich Gauss is the sum of all primitive roots.

The values of ${\displaystyle \mu (n)}$ for the first 50 positive numbers are

The first 50 values of the function are plotted below:

Larger values can be checked in: