## The Dirichlet Product of Arithmetic Functions

Dirichlet products or Dirichlet convolutions are any sum of the formwhere andare arithmetical functions. We can writeor

Examples:

wherefor all

Dirichlet products are the number theory analogue of the convolution theorem in calculus. Dirichlet products are commutative and associative, so for any arithmetical functionswe have:

Theorem

1.(commutativity)

2.(associativity)

Proof:

Writing the summation as above whereandvary over all integers whose product ismakes the commutative property self evident.

To prove the associative property letand consider

in the same way if we letand considerwe are led to the same formula for hencewhich means that Dirichlet multiplication is associative.

The Dirichlet product has certain very useful properties:

Ifandare multiplicative so thatandthen so is their Dirichlet productso that

Also ifandare multiplicative then so is

Ifis multiplicative then so is the Dirichlet inverse of