On the Inversion Polynomial for Dedekind Sums

TL;DR

This paper introduces the inversion polynomial for Dedekind sums to analyze their distribution, explores its properties and connections to Kloosterman sums, and improves bounds on Dedekind sum values.

Contribution

It defines the inversion polynomial for Dedekind sums, studies its properties, and establishes new bounds and conjectures related to Dedekind sums and their symmetries.

Findings

Properties of the inversion polynomial are established.

Connections between Kloosterman sums and the polynomial are demonstrated.

Bounds on the second highest Dedekind sum value are improved.

Abstract

We introduce the inversion polynomial for Dedekind sums $f_b(x)=\sum x^{\operatorname{inv}(a,b)}$ to study the number of $s(a,b)$ which have the same value for given $b$. We prove several properties of this polynomial and present some conjectures. We also introduce connections between Kloosterman sums and the inversion polynomial evaluated at particular roots of unity. Finally, we improve on previously known bounds for the second highest value of the Dedekind sum and provide a conjecture for a possible generalization. Lastly, we include a new restriction on equal Dedekind sums based on the reciprocity formula.