Class number zeta function of imaginary quadratic fields

TL;DR

This paper introduces a new zeta function that counts imaginary quadratic fields by their class numbers, proving its rationality and deriving bounds on the number of such fields with prime class numbers, using dynamical systems and representation theory.

Contribution

It defines a novel class number zeta function for imaginary quadratic fields and proves its rationality based on roots of unity, linking number theory with dynamical systems and operator theory.

Findings

The zeta function is rational and depends only on roots of unity.

A lower bound of 2p for the number of fields with prime class number p.

Method involves studying periodic points in a dynamical system related to Drinfeld modules.

Abstract

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a lower bound $2p$ for the number of imaginary quadratic fields of the prime class number $p$. Our method is based on the study of periodic points of a dynamical system arising in the representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space.