Math 2.0

Number theory Theorem

Dirichlet convolution of multiplicative functions is multiplicative

Builds on Dirichlet convolutionMultiplicative function

Statement

Let ff and gg be multiplicative functions. Then their Dirichlet convolution f∗gf * g is a multiplicative function.

Remarks

Let mm and nn be coprime. The proof uses that each divisor of mnmn is a unique product of a divisor of mm and a divisor of nn. With f=idk ⁣:n↦nkf = \mathrm{id}_k \colon n \mapsto n^k and g=𝟏g = \mathbf{1}, the theorem gives the multiplicativity of the divisor function, since σk=idk∗𝟏\sigma_k = \mathrm{id}_k * \mathbf{1}. The Dirichlet inverse of a multiplicative function is also multiplicative.

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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…definitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionDirichlet convolution, definition, no evidence levelDirichlet convolutiondefinitionCoprime integers, definition, no evidence levelCoprime integersdefinitionMultiplicative function, definition, no evidence levelMultiplicative functi…definitionDirichlet convolution of multiplicative functions is multiplicative, statement, no evidence levelDirichlet convolution…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. Divisors of a natural number definition
  4. Greatest common divisor definition
  5. Dirichlet convolution definition
  6. Coprime integers definition
  7. Multiplicative function definition
  8. Dirichlet convolution of multiplicative functions is multiplicative statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet_convolution, Properties
About this page
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Kindstatement, theorem
WritingW0 stub
MSC11A25
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UsesDirichlet convolution, Multiplicative function

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