Math 2.0

Number theory Arithmetic functions Theorem

Möbius inversion formula

Builds on Möbius functionThe Möbius function is the inverse of the constant function 1Dirichlet convolution

Statement

Let ff and gg be arithmetic functions. Then g(n)=∑d∣nf(d)g(n) = \sum_{d \mid n} f(d) for all positive integers nn if and only if f(n)=∑d∣nμ(d) g ⁣(nd)f(n) = \sum_{d \mid n} \mu(d) \, g\!\left(\frac{n}{d}\right) for all positive integers nn. Here μ\mu is the Möbius function.

Remarks

In terms of the Dirichlet convolution, the statement is: g=f∗𝟏g = f * \mathbf{1} if and only if f=g∗μf = g * \mu. The proof uses μ∗𝟏=ε\mu * \mathbf{1} = \varepsilon and the associativity of ∗*. For example, the sum of the totient over the divisors gives ∑d∣nφ(d)=n\sum_{d \mid n} \varphi(d) = n. So φ(n)=∑d∣nμ(d) n/d\varphi(n) = \sum_{d \mid n} \mu(d) \, n/d.

History

August Ferdinand Möbius introduced the formula into number theory in 1832.

Reading path

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionDivisors of a natural number, definition, no evidence levelDivisors of a natural…definitionPrime number, definition, no evidence levelPrime numberdefinitionDirichlet convolution, definition, no evidence levelDirichlet convolutiondefinitionNumber of distinct prime factors, definition, no evidence levelNumber of distinct pr…definitionSquare-free integer, definition, no evidence levelSquare-free integerdefinitionMöbius function, definition, no evidence levelMöbius functiondefinitionThe Möbius function is the inverse of the constant function 1, statement, no evidence levelThe Möbius function i…statementMöbius inversion formula, statement, no evidence levelMöbius inversion form…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.
  • no evidence level
  1. Arithmetic function definition
  2. Divisibility definition
  3. Divisors of a natural number definition
  4. Prime number definition
  5. Dirichlet convolution definition
  6. Number of distinct prime factors definition
  7. Square-free integer definition
  8. Möbius function definition
  9. The Möbius function is the inverse of the constant function 1 statement
  10. Möbius inversion formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula, Statement of the formula
About this page
Tag07Q4TBTFVB
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Kindstatement, theorem
WritingW1 stated
MSC11A25
StatusThis copy of the site has no build data. Run m2 status 07Q4TBTFVB.
UsesMöbius function, The Möbius function is the inverse of the constant function 1, Dirichlet convolution
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