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Reading the interesting paper Honest Bernoulli excursions by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known.

In the paper the authors consider a random walk on the integers $\mathbb Z$ with a particle starting at 0 and moving left or right with the same probability of $\frac 1 2$. Further, they let $T$ be the time of first return to zero , and $M_T$ the maximum distance from $0$ reached by the walk up to time $T$.

Their main result says that the probability $$P(M_T \leq y \sqrt{\pi n} \ | \ T=2n ) = F(y)+ O(n^{-\frac12})$$ is uniformly in $y$, with distribution function $F(y)$ (defined on $[0,\infty)$ given by $$F(y) = \frac{4\pi}{y^3} \sum_{j=1}^\infty j^2 exp(-\pi j^2 / y^2)$$

The relation with Riemann's completed zeta function comes now from the observation that the "Mellin transform of the limiting measure $F$" gives 2 times the completed Riemann zeta function: $$\int_0^\infty y^s F(dy) = 2 \xi(s)$$

Recall that $\xi(s) = \Gamma(\frac s2+1)(s-1)\pi^{-s/2}\zeta(s)$ (with the usual notations).

My question is now whether similar probabilistic interpretations are known for other Dedekind zeta functions. A first guess would be to look at random walks on the ring of integers $\mathcal O _K$ of a number field $K$, or on some other appropriate spaces like certain (signed) integral ideals of $\mathcal O _K$.

Is anything known in this direction? Any idea or reference would be highly appreciated. (I should say that my background in probability theory tends to zero, unfortunately.)

EDIT: As I already pointed this out in the comments: What I am really interested about in some sense is the question (very vaguely speaking) whether the above procedure can be generalized to give a probabilistic interpretation to Hecke's method of expressing Dedekind zeta functions (Hecke did this of course more generally for Hecke L-functions) in terms of Mellin transforms of appropriate $\vartheta$-functions. (I have Neukirch's presentation of Hecke's work in his number theory book in mind.)

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# Is there a probabilistic interpretation of Dedekind zeta functions?

Reading the interesting paper Honest Bernoulli excursions by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known.

In the paper the authors consider a random walk on the integers $\mathbb Z$ with a particle starting at 0 and moving left or right with the same probability of $\frac 1 2$. Further, they let $T$ be the time of first return to zero , and $M_T$ the maximum distance from $0$ reached by the walk up to time $T$.

Their main result says that the probability $$P(M_T \leq y \sqrt{\pi n} \ | \ T=2n ) = F(y)+ O(n^{-\frac12})$$ is uniformly in $y$, with distribution function $F(y)$ (defined on $[0,\infty)$ given by $$F(y) = \frac{4\pi}{y^3} \sum_{j=1}^\infty j^2 exp(-\pi j^2 / y^2)$$

The relation with Riemann's completed zeta function comes now from the observation that the "Mellin transform of the limiting measure $F$" gives 2 times the completed Riemann zeta function: $$\int_0^\infty y^s F(dy) = 2 \xi(s)$$

Recall that $\xi(s) = \Gamma(\frac s2+1)(s-1)\pi^{-s/2}\zeta(s)$ (with the usual notations).

My question is now whether similar probabilistic interpretations are known for other Dedekind zeta functions. A first guess would be to look at random walks on the ring of integers $\mathcal O _K$ of a number field $K$, or on some other appropriate spaces like certain (signed) integral ideals of $\mathcal O _K$.

Is anything known in this direction? Any idea or reference would be highly appreciated. (I should say that my background in probability theory tends to zero, unfortunately.)