I've been studying [1] because I was interested in his ideas on the zeta function. I'll define it here (c.f. p. 31):

The Kullback-Leibler distance is defined as 
$$
K(w)=\int q(x)f(x, w)dx\quad
f(x,w)=\log\frac{q(x)}{p(x|w)}
$$

For a given triple $(p, q, \varphi)$, where $p(x|w)$ is a statistical model (a p.d.f. at parameter $w$), $q(x)$ is a true probability distribution, and $\varphi(w)$ is an $\textit{a priori}$ probability density function with compact support, its zeta function for $z\in\mathbb{C}$
$$
\zeta(z)=\int K(w)^z\varphi(w)dw
$$
($K$ is generalized to any positive analytic function). I believe he is serious that is within the family of zeta functions because he ends the relevant chapter on the derivation and properties of the Riemann zeta function ([1], p. 132). However I understand there to be a criteria for (arithmetic) zeta functions [2]:

1. **Algebraicity**
$$
L(s)=\sum a_nn^{-s},\quad a_n\in\mathbb{Z}
$$
2. **Euler product**
$$
L(s)=\prod\phi_p(p^{-s})\quad\phi_p(X)\text{ is rational with bounded degree}
$$
3. **Functional Equation:** For some $h>0$ specific to the zeta function, the following holds for $z\in\mathbb{C}$: there exists a 'gamma factor' $\gamma(z)$ specific to the L-function
$$
\gamma(h-z)\zeta(h-z)=\gamma(z)\zeta(z)
$$
4. **Special values** (see cited paper)
$$\S$$

**Question 1**: In the interest of studying the zeta function of statistical models rigorously, has there been any effort to expand Zagier's (or similar) criteria to non-arithmetic zeta functions?

**Question 2**: Is there any research since this publication on the zeta function of statistical models? It seems like this work is not studied widely.

$$\S$$

[1]: Watanabe, S. *Algebraic Geometry and Statistical Learning Theory*

[2]: Zagier, D. https://www.euro-math-soc.eu/ECM/ECM1992.2/Main/ECM1992.2.0497.0512.pdf