Dedekind Sums with Even Denominators

TL;DR

This paper investigates the possible values of normalized Dedekind sums with even denominators, confirming conjectured restrictions in several cases and advancing understanding of their range.

Contribution

It verifies Girstmair's conjecture for specific classes of denominators and develops methods towards a general proof.

Findings

Confirmed conjecture for even denominators and certain square divisors.

Identified local restrictions on Dedekind sum values based on denominator properties.

Used computational methods to verify the conjecture for denominators up to 200.

Abstract

Let $S(a,b)$ denote the normalized Dedekind sum. We study the range of possible values for $S(a,b)=\frac{k}{q}$ with $\gcd(k,q)=1$. Girstmair proved local restrictions on $k$ depending on $q\pmod{12}$ and whether $q$ is a square and conjectured that these are the only restrictions possible. We verify the conjecture in the cases $q$ even, $q$ a square divisible by $3$ or $5$, and $2\le q\le 200$ (the latter by computer), and provide progress towards a general approach.