The **Dirichlet convolution** is a binary operation defined for arithmetic functions; it is of importance in number theory.

See also:

IfSome general properties of this operation include:

- If both
*f*and*g*are multiplicative, then so is*f***g*. (Note however that the convolution of two*completely*multiplicative functions need not be completely multiplicative.) -
*f***g*=*g***f*(commutativity) - (
*f***g*) **h*=*f** (*g***h*) (associativity) -
*f** (*g*+*h*) =*f***g*+*f***h*(distributivity) -
*f** ε = ε **f*=*f*, where ε is the function defined by ε(*n*) = 1 if*n*= 1 and ε(*n*) = 0 if*n*> 1. - To every multiplicative
*f*there exists a multiplicative*g*such that*f***g*= ε.

Furthermore, the multiplicative functions with convolution form an abelian group with identity element ε. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

If *f* is an arithmetic function, one defines its **L-series** by