Coh zeta functions for inert quadratic orders

TL;DR

This paper investigates the Coh zeta function for inert quadratic orders, proposing a conjecture linking it to $t$-deformed Bressoud $q$-series and providing explicit formulas to support this connection.

Contribution

It introduces a new method using M"obius inversion on posets and derives explicit formulas for the finitized Coh zeta functions, advancing understanding of their structure.

Findings

Conjecture relating Coh zeta functions to $t$-deformed Bressoud $q$-series.

Explicit formulas for the simplest order's finitized Coh zeta function.

Verification of the conjecture at $t=1$ for all orders in the family.

Abstract

We study the Coh zeta function for a family of inert quadratic orders, which we conjecture to be given by $t$-deformed Bressoud $q$-series. This completes a trilogy connecting the zeta functions of ramified and split quadratic orders to the classical Andrews--Gordon and Bressoud identities, respectively. We provide strong evidence for this conjecture by deriving the first explicit formulas for the finitized Coh zeta function of the simplest order in the family, and for the $t=1$ specialization of the finitized Coh zeta functions for all orders in the family. Our primary tool is a new method based on M\"obius inversion on posets.