
TL;DR
This paper investigates the possible numerator values of Dedekind sums for a fixed denominator, establishing a residue class structure that simplifies their classification and providing computational results for denominators up to 50.
Contribution
It proves that numerators of Dedekind sums form complete residue classes modulo a specific number, reducing the problem to finitely many residue class representatives and computing these for denominators up to 50.
Findings
Numerators form complete residue classes modulo (q^2-1)q.
All possible numerators for 2 ≤ q ≤ 50 are determined.
A conjecture on all possible Dedekind sum values is proposed.
Abstract
Let , where denotes the classical Dedekind sum. For a given denominator , we study the numerators of the values , , of Dedekind sums . Our main result says that if is such a numerator, then the whole residue class of modulo consists of numerators of this kind. This fact reduces the task of finding all possible numerators to that of finding representatives for finitely many residue classes modulo . By means of the proof of this result we have determined all possible numerators for , the case being trivial. The result of this search suggests a conjecture about all possible values , , of Dedekind sums for an arbitrary .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Coding theory and cryptography
