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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:

  1. First, anything that's known about it at all - if it has a name, if it's well understood, references about it, etc.
  2. If there exist "standard" ways to extend it to functions defined at zero, perhaps of a real variable.
  3. 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.
  4. Same as above, but where Dirichlet convolution maps to pointwise multiplication instead.
  5. Same as above, but where both Dirichlet and Cauchy convolution map to pointwise something, and pointwise addition is still preserved.
  6. 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.
  7. 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.
  8. A theory of "generating functions" related to the above.
  9. 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.

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    $\begingroup$ I don't understand why what you've already written isn't an answer to most of these questions. As you already know, sending a sequence $a_1, a_2, \dots$ to the corresponding formal power series $\sum a_i x^i$ turns Cauchy convolution into "pointwise" multiplication, and sending it to the formal Dirichlet series $\sum \frac{a_i}{i^s}$ turns Dirichlet convolution into "pointwise" multiplication. The quotation marks can be removed if you impose growth conditions. In what way is this not an answer to #3, #4, and #8? $\endgroup$ – Qiaochu Yuan Aug 3 '15 at 4:35
  • $\begingroup$ Because I want an isomorphism respecting all three operations. Given two formal Dirichlet series, how do I get a new Dirichlet series with the Cauchy convolution of their coefficients? $\endgroup$ – Mike Battaglia Aug 3 '15 at 4:41
  • $\begingroup$ Other than, I guess, the trivial way where you can convert the Dirichlet series back to an arithmetic function by integrating, and then do your convolution, and then re-attach back to the Dirichlet series. $\endgroup$ – Mike Battaglia Aug 3 '15 at 4:44
  • $\begingroup$ I don't know that you have anything more special than two bilinear operators (convolutions) over (the countable power of) a group. If you have some equational relations involving the two convolutions, you might be looking at something special which has been studied before. There are forms (cf. Movsisyan) of "hyperdistributivity", where certain distributive relations are presumed to hold, but your convolutions may not fit that. Gerhard "Consider The Free Such Algebra?" Paseman, 2015.08.03 $\endgroup$ – Gerhard Paseman Aug 4 '15 at 0:52
  • $\begingroup$ If you restrict the entire domain to delta functions and throw away addition, then Dirichlet convolution distributes over ordinary convolution. I'm trying to figure out if that extends in any meaningful way to the rest of the domain, though. $\endgroup$ – Mike Battaglia Aug 4 '15 at 7:30
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Ack - Dirichlet convolution does NOT distribute over ordinary convolution. I went wrong somewhere way in the beginning of this process and need to backtrack.

See below, where sequences are indexed starting from 1:

$ [2,0,0,0,...] \ast_D \left( [1,0,0,0,...] \ast_C [1,0,0,0,...] \right) \\ = [2,0,0,0,...] \ast_D [0,1,0,0,...] \\ = [0,2,0,0,...] $

but

$ [2,0,0,0,...] \ast_D \left( [1,0,0,0,...] \ast_C [1,0,0,0,...] \right) \\ = \left( [2,0,0,0,...] \ast_D [1,0,0,0,...] \right) \ast_C \left( [2,0,0,0,...] \ast_D [1,0,0,0,...] \right)\\ = [2,0,0,0,...] \ast_c [2,0,0,0,...] \\ = [0,4,0,0,...] $

I'm not sure if these two operations interact in any kind of "nice" way at all, but at least sanity has been restored.

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