Shintani's invariant via cyclic quantum dilogarithm

TL;DR

This paper introduces a new formulation of Shintani's invariant using the cyclic quantum dilogarithm, offering fresh insights into its arithmetic properties and connecting it to quantum algebraic structures.

Contribution

It presents the first formulation of Shintani's invariant via the cyclic quantum dilogarithm, extending previous $q$-Pochhammer symbol approaches.

Findings

Cyclic quantum dilogarithm naturally encodes Shintani's invariant.

Provides a new perspective linking arithmetic invariants and quantum functions.

Lays groundwork for further detailed proofs and applications.

Abstract

We formulate Shintani's invariant in terms of the cyclic quantum dilogarithm. Building on earlier results that expressed Shintani's invariant using the $q$-Pochhammer symbol, we show how the cyclic quantum dilogarithm naturally arises in this context, providing new perspectives on the arithmetic significance of Shintani's construction. The present note is an announcement; a full account with complete proofs will appear in a forthcoming paper.