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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?

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[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

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