We define the Dirichlet convolution of two arithmetic functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell series of $a$ to be $$ a_p(x)=\sum_{n=0}^{\infty}a(p^n)x^n. $$ Both of these definitions are taken from Apostol's Introduction to Analytic Number Theory. It is easy to show that if $c=a*b$, then $c_p(x)=a_p(x)b_p(x)$ for all primes $p$.
There are some pretty obvious cosmetic similarities here between the definition of Dirichlet convolution and this result with Bell series and the usual notion of convolution in analysis and how the Fourier transform interacts with it ($\widehat{f*g}=\hat{f}\cdot\hat{g}$). Are these similarities actually significant?
More specifically, can Dirichlet convolution be viewed somehow as the usual convolution of two functions on some group with a measure? Are Bell series related in any way to Fourier transforms or Fourier coefficients, or is it just nice and convenient to have results about different types of products interacting well with each other?
