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!

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