PlanetMath: Möbius function

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (203)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Möbius function

(Definition)

The Möbius function of number theory is the function $\mu:\mathbb{Z}^+\to\{-1,0,1\}$ defined by

$\begin{displaymath} \mu (n) = \begin{cases} 1, &\text{if $n=1$}\ 0, &\text{if ... ...\cdots p_r$, where the $p_i$\ are distinct primes.} \end{cases}\end{displaymath}$

In other words, $\mu (n) = 0$ if is not a square-free integer, while $\mu (n) = (-1)^r$ if is square-free with prime factors. The function $\mu$ is a multiplicative function, and obeys the identity

$\begin{displaymath} \sum_{d \vert n} \mu(d) = \begin{cases} 1 & \text{if $n = 1$}\ 0 & \text{if $n > 1$} \end{cases}\end{displaymath}$

where

runs through the positive divisors of

Anyone with an account can edit this entry. Please help improve it!

"Möbius function" is owned by mps. [ full author list (3) | owner history (2) ]

(view preamble)

See Also: square-free number, sum of $\frac{\mu(n)}{n}$ , Möbius inversion, convolution method

Other names:

Moebius function

Keywords:

number theory

Attachments:: uniqueness of Moebius function (Definition) by mathcam
sum of $\frac{\mu(n)}{n}$ (Result) by mathcam
table of values of the Möbius function and the Mertens function (Data Structure) by PrimeFan

Log in to rate this entry.
(view current ratings)

Cross-references: divisors, positive, identity, multiplicative function, prime factors, integer, square-free, function, number theory
There are 23 references to this entry.

This is version 6 of Möbius function, born on 2001-10-16, modified 2005-07-26.
Object id is 253, canonical name is MoebiusFunction.
Accessed 8970 times total.

Classification:

AMS MSC:

11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

Interact