There is a connection between the product of the Bell series of two functions and their Dirichlet convolution.
Theorem
Let
and
be arithmetical functions and let
then for every prime
we have
Proof: Since the divisors of
are
we have
The last sum is the Cauchy product of the sequences
and
Examples:
so the Bell series of
modulois
so the Bell series of
modulois 
Bell series can be used to investigate the properties of arithmetical functions. If
where 
and
thenis multiplicative and it's Bell series modulois
Hence
which implies
or