# Arithmetic function

Revision as of 05:03, 15 June 2009 by imported>Jitse Niesen (remove "See also" section because it is subsumed in the "Related Articles" subpage; perhaps the "Examples" section should go for the same reason, though it may be better to flesh it out a bit)

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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

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

### Multiplicative functions

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

**Totally multiplicative**if for all*m*and*n*.**Multiplicative**if whenever*m*and*n*are coprime.

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

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