EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference.
I keep stumbling on the same algebraic structure, and I have no clue how to understand or characterize it at all. It's basically the merger of the Dirichlet ring and the ordinary convolution ring. I've run into some interesting questions from studying it, and I'm hoping someone can shed some light on it.
Consider the structure $\langle \Bbb C^{\Bbb N^+}, \ast_D, \ast_C,+\rangle$, where $\ast_D$ is Dirichlet convolution, $\ast_C$ is ordinary/Cauchy convolution, and $\Bbb N^+$ is the naturals without zero.
The three operations in this structure are commutative, "pairwise-distributive," and totally ordered in a "hierarchy" of distributivity. In other words: $\ast_D$ distributes over $\ast_C$, $\ast_C$ distributes over $+$, and $\ast_D$ distributes over $+$.
Now, assuming $\delta_n$ is a Kroenecker delta spiking at n, one useful equation to understand the mechanics of this structure is $\delta_a \ast_D (\delta_b \ast_C (\delta_c + \delta_d)) = \delta_{a(b+c)} + \delta_{a(b+d)}$. From this perspective, it's easy to envision ways to extend this structure to a larger domain, save for some difficulty in treating $\delta_0$ and functions defined at 0 in general.
Here's what I'm trying to figure out:
- First, anything that's known about it at all - if it has a name, if it's well understood, references about it, etc.
- If there exist "standard" ways to extend it to functions defined at zero, perhaps of a real variable.
- If there exists an injective "homomorphism," possibly for an enlarged domain, in which the image is some other set of complex-valued functions, pointwise addition maps to pointwise addition, and the Cauchy convolution maps to pointwise multiplication.
- Same as above, but where Dirichlet convolution maps to pointwise multiplication instead.
- Same as above, but where both Dirichlet and Cauchy convolution map to pointwise something, and pointwise addition is still preserved.
- If the isomorphisms I'm looking for in #3-5, can be described by some useful "transform," similar to the Laplace or Mellin or Dirichlet or Z-transforms, etc.
- If there exists a formal "series" in some variable, with two operations defined on it, such that applying these operations to the series gives you a new series with the Cauchy/Dirichlet convolution of the coefficients.
- A theory of "generating functions" related to the above.
- Any of the above, but where the foundations are tweaked slightly to involve functions of a real variable, or tempered distributions, or anything that makes this more simple.
Hopefully the motivation for some of these questions is clear - if you consider either convolution individually, there is a pretty rich "universe" of concepts easily associated with it.
- Ordinary convolution is easily seen to be related to power series, ordinary generating functions, the Z-transform, the Laplace/Fourier transforms, "shift-invariance,""translations,""additive" things, etc.
- Dirichlet convolution quickly turns into stuff about Dirichlet series, Dirichlet generating functions, the Dirichlet and Mellin transforms, "scale-invariance,""dilations,""multiplicative" things, etc.
I'm trying to understand if there's a similarly rich theory for understanding what happens when you mix both operations together. Things related to this structure have so far turned up for me in number theory, digital signal processing, and even some work I'm doing in music theory (which, the way we're doing it, is really like "applied number theory"). It would be helpful to get some insight.
