The Dirichlet Product of Arithmetic Functions
Dirichlet products or Dirichlet convolutions are any sum of the form
where 
and
are arithmetical functions. We can write
or
Examples:
where
for all
Dirichlet products are the number theory analogue of the convolution theorem in calculus. Dirichlet products are commutative and associative, so for any arithmetical functions
we have:
Theorem
1.
(commutativity)
2.
(associativity)
Proof:
Writing the summation as above where
and
vary over all integers whose product is
makes the commutative property self evident.
To prove the associative property let
and consider
in the same way if we let
and consider
we are led to the same formula for
hence
which means that Dirichlet multiplication is associative.
The Dirichlet product has certain very useful properties:
Ifandare multiplicative so that
and
then so is their Dirichlet productso that
Also ifand
are multiplicative then so is
Ifis multiplicative then so is
the Dirichlet inverse of