Möbius function

Builds on Square-free integer Number of distinct prime factors

Definition

The Möbius function $\mu$ is an arithmetic function. For each positive integer $n$ , it is given by $\mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is squarefree}, \\ 0 & \text{otherwise}. \end{cases}$ Here $\omega(n)$ is the number of distinct prime factors of $n$ . The word squarefree has the meaning of the definition of a squarefree integer.

Remarks

So $\mu(1) = 1$ , $\mu(p) = -1$ for each prime $p$ , and $\mu(n) = 0$ when $p^2 \mid n$ for some prime $p$ . For example, $\mu(6) = 1$ , $\mu(30) = -1$ and $\mu(12) = 0$ . The function $\mu$ is multiplicative, and it is the inverse of the constant function 1 for the Dirichlet convolution.

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Arithmetic function definition
Divisibility definition
Prime number definition
Number of distinct prime factors definition
Square-free integer definition
Möbius function definition

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5 pages build on this one.

Tag	`0C8QENHYZW`
URI	`m2://0C8QENHYZW`
Kind	definition
Writing	W1 stated
MSC	11A25
Defines	Möbius function
Uses	Square-free integer, Number of distinct prime factors

Möbius function

Definition

Remarks

Reading path

Where this leads

Further reading

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