Bell Series and Dirichlet Multiplication

There is a connection between the product of the Bell series of two functions and their Dirichlet convolution.


Letandbe arithmetical functions and letthen for every primewe have

Proof: Since the divisors ofarewe have

The last sum is the Cauchy product of the sequencesand


so the Bell series ofmodulois

so the Bell series ofmodulois

Bell series can be used to investigate the properties of arithmetical functions. Ifwhere andthenis multiplicative and it's Bell series modulois

Hencewhich impliesor