Multiplicative function: Difference between revisions

Revision as of 12:18, 19 October 2021

Revision as of 14:42, 19 October 2021

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== Convolution ==

If ''f'' and ''g'' are two multiplicative functions, one defines a new multiplicative function ''<math>f'' * ''g''</math>, the ''[[Dirichlet convolution]]'' of ''f'' and ''g'', by

<math display="block"> (f \, * \, g)(n) = \sum_{d|n} f(d) \, g \left( \frac{n}{d} \right)</math>

where the sum extends over all positive divisors ''d'' of ''n''.

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Relations among the multiplicative functions discussed above include:

* ''μ''<math>\mu * 1 = ''ε''\varepsilon</math> (the [[Möbius inversion formula]])

* <math>(''μ'' \mu \operatorname{Id_''k''}_k) * \operatorname{Id<sub>''k''}_k = \varepsilon</submath> = ''ε'' (generalized Möbius inversion)

* <math>\varphi</math> * 1 = \operatorname{Id}</math>

* ''<math>d'' = 1 * 1</math>

* ''σ''<math>\sigma = \operatorname{Id} * 1 = <math>\varphi</math> * ''d''</math>

* <math>\sigma_k = \operatorname{Id}_k * 1</math>

* ''σ''_''k'' = Id_''k'' * 1

* <math>\operatorname{Id} = <math>\varphi</math> * 1 = ''σ''\sigma * ''μ''\mu</math>

* <math>\operatorname{Id}_k = \sigma_k * \mu</math>

* Id_''k'' = ''σ''_''k'' * ''μ''

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the [[Dirichlet ring]].