Proposition: Sum of Möbius Function Over Divisors

For any natural number $n\ge 1$, the sum of the Möbius function over the divisors of $n$ equals $0$ unless $n=1$. Only for this special case the sum equals $1.$ Using the "Iverson notation for sums":, this can be written as

$$\sum_{d\mid n}\mu(d)=[n=1].$$

Proofs: 1

Proofs: 1 2

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