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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

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

It was introduced by August Ferdinand 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)

| | | | | created: 2019-03-17 19:33:16 | modified: 2019-04-06 08:43:20 | by: bookofproofs | references: [701], [1272]

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Bibliography (further reading)

[1272] Landau, Edmund: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

[701] Scheid Harald: “Zahlentheorie”, Spektrum Akademischer Verlag, 2003, 3. Auflage