# 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.)

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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The limiting measure is just the distribution of the maximum of a Brownian motion. I also did this calculation in a recent answer on math.stackexchange. math.stackexchange.com/questions/38642/…. I'm not aware of links with other types of zeta functions though. – George Lowther Jun 11 '11 at 14:01
Dear George, thank you very much for the clarification! – user5831 Jun 11 '11 at 15:07
You can certainly construct analogues of the zeta distribution that take values in other number fields by means of a Dedekind zeta function. These have similar properties to the standard zeta distribution, including the "independence of prime factors" property. – Simon Lyons Jun 11 '11 at 17:32
Dear Simon, is it possible to argue that these distributions may arise from a random walk on an appropriate space? The distribution mentioned above is related to the classical theta function and for general number fields we know by the work of Hecke how to write down "generalized theta functions" whose Mellin transform give rise to completed Dedekind zeta functions. In some sense I am asking if the work of Hecke has a probabilistic interpretation? – user5831 Jun 12 '11 at 14:31
I think there may be issues related to positivity of the generalised theta functions. Biane, Pitman and Yor encountered a similar problem when they considered more general L functions. See section six of this paper: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.3091 – Simon Lyons Jun 12 '11 at 22:06