# Functional Equation of Zeta Function on Statistical Model

I’ve been studying  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 (, p. 132). However I understand there to be a criteria for (arithmetic) zeta functions :

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?

$$\S$$

: Watanabe, S. Algebraic Geometry and Statistical Learning Theory

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