# Generating functions in countable commutative monoids

Let $$f: \mathbb{N}_0 \rightarrow \mathbb{C}$$ be a function. The power series of $$f$$ can be viewed as the function $$\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$$ where $$q \in \mathbb{C}$$ such that the aforementioned (non-formal) series is convergent. Note that for all $$q \in \mathbb{C}$$, $$n \mapsto q^n$$ is a monoid homomorphism from $$(\mathbb{N}_0, +)$$ to $$(\mathbb{C}, \cdot)$$. Also, if the series $$\sum_{n \in \mathbb{N}_0}^{} f(n)q^n$$ converges to zero on some non-empty open subset of $$\mathbb{C}$$, then $$f$$ must be the zero function.

On the other hand, let $$f: \mathbb{N} \rightarrow \mathbb{C}$$ be a function. The Dirichlet series of $$f$$ can be viewed as the function $$\mathscr{D}_f : s \mapsto \sum_{n \in \mathbb{N}}^{} f(n)n^{-s}$$ where $$s \in \mathbb{C}$$ such that the aforementioned (non-formal) series is convergent. Note that for all $$s \in \mathbb{C}$$, $$n \mapsto n^{-s}$$ is a monoid homomorphism from $$(\mathbb{N}, \cdot)$$ to $$(\mathbb{C}, \cdot)$$. Also, if the series $$\sum_{n \in \mathbb{N}}^{} f(n)n^{-s}$$ converges to zero on some open subset of $$\mathbb{C}$$, then $$f$$ must be the zero function.

Let $$(M, \oplus)$$ be a countably infinite commutative monoid and let $$f: M \rightarrow \mathbb{C}$$ be a mapping. When $$M$$ is well-ordered, say by $$\leq$$, such that $$(M, \leq)$$ is a locally-finite poset, we will define the series $$\sum_{m \in M} f(m)$$ as the limit of partial sums of $$f(m)$$ starting at the least element of $$M$$. Note that a well-ordering $$\leq$$ ensures that all elements of $$M$$ are arranged in a countable non-decreasing chain.

In the spirit of the observations mentioned above, we can introduce the following definition:

Definition. A monoid with generating functions is a commutative monoid $$(M, \oplus)$$ for which there exists a well order $$\leq$$ on $$M$$ such that $$(M, \leq)$$ is a locally-finite poset (implying that $$M$$ is necessarily countable) and for which there exists a non-empty subset $$\Phi \subseteq \mathrm{Hom}((M,\oplus), (\mathbb{C}, \cdot))$$ such that for every mapping $$f: M \rightarrow \mathbb{C}$$ and non-empty subset $$U \subseteq \Phi$$ open in the topology $$\tau_{\Phi}$$ of pointwise convergence on $$\Phi$$ induced by the usual topology on $$\mathbb{C}$$, we have the implication:

$$\left (\forall \phi \in U : \sum_{m \in M}^{} f(m) \phi(m) = 0 \right ) \Rightarrow f=0.$$

Every such ordered quadruple $$(M, \oplus, \leq, \Phi)$$ will be called a generating function system (briefly, a GFS). We will only work in the topology $$\tau_{\Phi}$$ when considering a GFS $$(M, \oplus, \leq, \Phi)$$.

Given a mapping $$f: M \rightarrow \mathbb{C}$$, we will define

$$\mathscr{C}(f)=\left \{ \phi \in \Phi : \sum_{m \in M}^{} f(m) \phi(m) \ \mathrm{converges} \right \}.$$

If $$\mathscr{C}(f) \neq \emptyset$$, we will call the resulting function $$\mathscr{G}_{f} : \mathscr{C}(f) \rightarrow \mathbb{C}$$ defined such that $$\mathscr{G}_{f}(\phi)=\sum_{m \in M}^{} f(m) \phi(m)$$ for all $$\phi \in \mathscr{C}(f)$$ the generating function of $$f$$.

A non-interesting example of a GFS would be a trivial monoid with its unique well order and homomorphism to $$(\mathbb{C}, \cdot)$$. My questions are the following:

Question 1 (main question):

Given a countable commutative monoid $$(M, \oplus)$$, a well order $$\leq$$ on $$M$$ such that $$(M, \leq)$$ is a locally-finite poset, and a non-empty subset $$\Phi \subseteq \mathrm{Hom}((M,\oplus), (\mathbb{C}, \cdot))$$, what is some sufficient condition on $$\Phi$$ for $$(M, \oplus, \leq, \Phi)$$ to be a GFS?

A plausible approach could be to find a sufficient condition for generating functions to be analytic on some non-empty subset of $$\mathbb{C}$$ and apply the identity theorem for analytic functions. This might be done by indexing $$\Phi$$ by complex numbers and requiring analycity of monoid homomorphisms. However, I am not sure how to deduce the all coefficients of the zero function must be zero.

Question 2:

It is clear that the power series of every function $$f: \mathbb{N}_0 \rightarrow \mathbb{C}$$ converges on a region of $$\mathbb{C}$$ that is of empty interior if and only if it is the region $$\left \{0 \right \}$$. Similarly, it is clear that the Dirichlet series of every function $$f: \mathbb{N} \rightarrow \mathbb{C}$$ converges on a region of $$\mathbb{C}$$ that is of empty interior if and only if it is the region $$\emptyset$$.

I am interested in whether or not the following proposition holds:

Proposition. Let $$(M, \oplus, \leq, \Phi)$$ be a GFS, $$e$$ the identity element of $$(M, \oplus)$$, and let $$f: M \rightarrow \mathbb{C}$$ be a mapping. If $$\mathrm{int}\mathscr{C}(f)=\emptyset$$, then $$\mathscr{C}(f) \in \left \{\emptyset, \left \{\phi_0 \right \} \right \}$$ where $$\phi_0$$ is the homomorphism defined such that $$\phi_0(m)=0$$ for all $$m \in M \setminus \left \{e \right \}$$.

NOTE:

I have posted a similar question on MSE, but decided to post this more comprehensive question. Anyhow, on MSE, @Lukas Heger has noted how a similar theory, namely, abstract analytic number theory, has been developed. Also, I mist thank @Sam Hopkins for bringing to my attention the notion of a total algebra of a monoid which is connected to what I am doing here.

Your help in addressing any of the two questions (though Question 1 is certainly of more importance to me) is greatly appreciated!

• Okay, I removed that comment. I noticed you also deleted your question from MSE. There was another comment there which suggested you look into abstract analytic number theory (en.wikipedia.org/wiki/Abstract_analytic_number_theory), as an already developed theory which may be similar to what you are trying to achieve here. This would be a more analytic flavored approach than my algebraic suggestion of the total algebra (en.wikipedia.org/wiki/Total_algebra). From the many corrections you've had to make here, you can see why using an existing theory might be preferable. Commented Feb 29 at 16:09
• If you're interested in convergence properties of power series and Dirichlet series, you might also want to look at "generalized Dirichlet series" (en.wikipedia.org/wiki/General_Dirichlet_series) which can incorporate both. Commented Feb 29 at 16:49
• I insist a bit: $\mathscr{C}(f)=\left \{ \phi \in \Phi : \sum_{m \in M}^{} f(m) \phi(m) < \infty \right \}.$ has really NO meaning, sorry. The reason is that all the partial sums take their values in $\mathbb{C}$. Commented Mar 5 at 12:53
• @DuchampGérardH.E. I agree, fixed. Commented Mar 5 at 16:20
• @DuchampGérardH.E. Is this clarification OK? Commented Mar 6 at 12:48