Is there a probabilistic interpretation of Dedekind zeta functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:33:30Z http://mathoverflow.net/feeds/question/67506 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67506/is-there-a-probabilistic-interpretation-of-dedekind-zeta-functions Is there a probabilistic interpretation of Dedekind zeta functions? BY 2011-06-11T13:36:38Z 2011-06-12T14:36:30Z <p>Reading the interesting paper <a href="http://www-stat.stanford.edu/~cgates/PERSI/papers/honest_bern.pdf" rel="nofollow">Honest Bernoulli excursions</a> by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known.</p> <p>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 $T$.</p> <p>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 $$F(y) = \frac{4\pi}{y^3} \sum_{j=1}^\infty j^2 exp(-\pi j^2 / y^2)$$</p> <p>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: $$\int_0^\infty y^s F(dy) = 2 \xi(s)$$ </p> <p>Recall that $\xi(s) = \Gamma(\frac s2+1)(s-1)\pi^{-s/2}\zeta(s)$ (with the usual notations).</p> <p>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 $\mathcal O _K$.</p> <p>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.)</p> <p>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.)</p>