Arithmetic function

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This editable Main Article is under development and subject to a disclaimer.

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In number theory, an arithmetic function is a function defined on the set of positive integers, usually with integer, real or complex values.

## Classes of arithmetic function[edit]

Arithmetic functions which have some connexion with the additive or multiplicative structure of the integers are of particular interest in number theory.

### Multiplicative functions[edit]

We define a function a(n) on positive integers to be

* Totally multiplicative if a(mn)=a(m)a(n) for all m and n.
* Multiplicative if a(mn)=a(m)a(n) whenever m and n are coprime.

The Dirichlet convolution of two arithmetic function a(n) and b(n) is defined as

a⋆b(n)=∑d∣na(d)b(n/d).

If a and b are multiplicative, so is their convolution.

## Examples[edit]

* Carmichael's lambda function
* A Dirichlet character
* Euler's totient function
* Jordan's totient function
* Möbius function