The Möbius function is the inverse of the constant function 1

Builds on Möbius function Dirichlet convolution

Statement

Let $\mathbf{1}$ be the arithmetic function with $\mathbf{1}(n) = 1$ for all $n$ , and let $\varepsilon$ be the function with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n > 1$ . Then $\mu * \mathbf{1} = \varepsilon$ , where $\mu$ is the Möbius function and $*$ is the Dirichlet convolution. That is, for each positive integer $n$ , $\sum_{d \mid n} \mu(d) = \begin{cases} 1 & \text{if } n = 1, \\ 0 & \text{if } n > 1. \end{cases}$

Remarks

The proof for $n > 1$ : let $k = \omega(n) \ge 1$ . Only squarefree divisors count, so the sum is $\sum_{j=0}^{k} \binom{k}{j} (-1)^j = (1 - 1)^k = 0$ . For example, the divisors of $12$ give $\mu(1) + \mu(2) + \mu(3) + \mu(6) = 1 - 1 - 1 + 1 = 0$ . This identity is the key step in the Möbius inversion formula.

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Arithmetic function definition
Divisibility definition
Divisors of a natural number definition
Prime number definition
Dirichlet convolution definition
Number of distinct prime factors definition
Square-free integer definition
Möbius function definition
The Möbius function is the inverse of the constant function 1 statement

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Möbius inversion formula

Tag	`0ASC73QEQT`
URI	`m2://0ASC73QEQT`
Kind	statement, theorem
Writing	W0 stub
MSC	11A25
Status	This copy of the site has no build data. Run `m2 status 0ASC73QEQT`.
Uses	Möbius function, Dirichlet convolution

The Möbius function is the inverse of the constant function 1

Statement

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