Definition: Möbius Function, Square-free

The Möbius Moebius) function $\mu:\mathbb N\to\{-1,0,1\}$ is an arithmetic function indictating if a natural number $n > 0$ is square-free, i.e. if in its canonical representation. $$ n=\prod_{i=1}^\infty p_i^{e_i}$$

all exponents $e_i$ are less or equal $1.$ More precisely, the Möbius function is defined by

$$\mu(n) := \begin{cases} 1 & \text{if } n=1\\ (-1)^{r} & \text{if } n \text{ is square-free, i.e. a product of }r\text{ distinct primes }\\ 0 & \text{else} \end{cases}\quad\quad\forall n > 0.$$

It was introduced by August Möbius (1790 - 1868) and plays a prominent role in number theory, as we will see later.


The $\mu$ function can be visualized using SageMath. If you click on the evaluate button, you will see the values of $\mu(n)$ for $n=1,\ldots,100.$ It is oscillating between the values $0$ $-1,$ and $+1.$

moebiuspoints= [(i, moebius(i)) for i in range(1,100)] list_plot(moebiuspoints)

Examples: 1
Explanations: 2
Lemmas: 3 4
Proofs: 5 6 7 8 9
Propositions: 10
Sections: 11
Theorems: 12

Thank you to the contributors under CC BY-SA 4.0!




  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927