Definition:Möbius Function

Definition

Let $n \in \Z_{>0}$, that is, a strictly positive integer.

The Möbius function is the function $\mu: \Z_{>0} \to \Z_{>0}$ defined as:

$\map \mu n = \begin{cases} 1 & : n = 1 \\ 0 & : \exists p \in \mathbb P: p^2 \divides n\\ \paren {-1}^k & : n = p_1 p_2 \ldots p_k: p_i \in \mathbb P \end{cases}$

That is:

\(\ds \map \mu 1\)

\(=\)

\(\ds 1\)

\(\ds \map \mu n\)

\(=\)

\(\ds 0\)

if $n$ has any divisor which is the square of a prime, that is $n$ is not square-free

\(\ds \map \mu n\)

\(=\)

\(\ds \paren {-1}^k\)

if $n$ has $k$ distinct prime divisors.

Examples

Example: $1$

$\map \mu 1 = 1$

Example: $2$

$\map \mu 2 = -1$

Example: $4$

$\map \mu 4 = 0$

Example: $6$

$\map \mu 6 = 1$

Also known as

Möbius function is also seen rendered as Moebius function in environments where rendering the umlaut is inconvenient.

Also see

• Möbius Function is Multiplicative

• Results about the Möbius function can be found here.

Source of Name

This entry was named for August Ferdinand Möbius.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 25 \beta$
• 1976: Tom M. Apostol: Introduction to Analytic Number Theory ... (previous) ... (next): $2.2$: The Möbius function $\map \mu n$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Möbius function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Möbius function