Markov processes on the adeles and Dedekind's zeta function

TL;DR

This paper constructs a Markov process on the adeles of an algebraic number field to provide a probabilistic interpretation of its Dedekind zeta function, extending previous work from rational numbers to general number fields.

Contribution

It introduces a new Markov process on the adeles of a number field and links it to the Dedekind zeta function, broadening the probabilistic approach to algebraic number theory.

Findings

Probabilistic interpretation of $\zeta_K(s)$ for $ e s>1$

Extension of Yasuda's work from $\Q$ to general number fields

Construction of an additive Markov process on the adeles

Abstract

Let $K$ be an algebraic number field. We construct an additive Markov process $X_t^{K_\mathbb A}$ on the ring of adeles $K_\mathbb A,$ whose coordinates $X_t^{(v)}$ are independent and use this process to give a probabilistic interpretation of the Dedekind zeta function $\zeta_K(s),$ for $\re s>1.$ This note extends a recent work of Yasuda [J. Theor. Probab. 23(3):748--769, 2010] where the case of the field $K=\Q$ of rational numbers was considered.