Math 2.0

Number theory Proposition

Multiplicativity of the Möbius function

Builds on Möbius functionMultiplicative function

Statement

The Möbius function μ\mu is a multiplicative function. That is, μ(1)=1\mu(1) = 1 and μ(mn)=μ(m)μ(n)\mu(mn) = \mu(m) \mu(n) for all coprime positive integers mm and nn.

Remarks

The function μ\mu is not completely multiplicative: μ(4)=0\mu(4) = 0, but μ(2)2=1\mu(2)^2 = 1. The proof: for coprime mm and nn, the product mnmn is squarefree exactly when mm and nn are both squarefree. Then ω(mn)=ω(m)+ω(n)\omega(mn) = \omega(m) + \omega(n).

Reading path

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionPrime number, definition, no evidence levelPrime numberdefinitionCoprime integers, definition, no evidence levelCoprime integersdefinitionNumber of distinct prime factors, definition, no evidence levelNumber of distinct pr…definitionSquare-free integer, definition, no evidence levelSquare-free integerdefinitionMultiplicative function, definition, no evidence levelMultiplicative functi…definitionMöbius function, definition, no evidence levelMöbius functiondefinitionMultiplicativity of the Möbius function, statement, no evidence levelMultiplicativity of t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Arithmetic function definition
  2. Divisibility definition
  3. Greatest common divisor definition
  4. Prime number definition
  5. Coprime integers definition
  6. Number of distinct prime factors definition
  7. Square-free integer definition
  8. Multiplicative function definition
  9. Möbius function definition
  10. Multiplicativity of the Möbius function statement

Further reading

  1. Web https://en.wikipedia.org/wiki/M%C3%B6bius_function, Properties
About this page
Tag0VB9G5B034
URIm2://0VB9G5B034
Kindstatement, proposition
WritingW0 stub
MSC11A25
StatusThis copy of the site has no build data. Run m2 status 0VB9G5B034.
UsesMöbius function, Multiplicative function

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