# Functional Equation of Zeta Function on Statistical Model

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? Particularly, any efforts to establish a functional equation?

$$\S$$

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