2016/10/18 by Kurt Girstmair, Girstmair, Kurt
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1610.05588
openalex publication_date 2016/10/18 · arxiv created 2016/10/27 · arxiv updated 2016/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let S(a,b)=12s(a,b), where s(a,b) denotes the classical Dedekind sum. For a given denominator q∈ \mathbb N, we study the numerators k∈\mathbb Z of the values k/q, (k,q)=1, of Dedekind sums S(a,b). Our main result says that if k is such a numerator, then the whole residue class of k modulo (q2-1)q consists of numerators of this kind. This fact reduces the task of finding all possible numerators k to that of finding representatives for finitely many residue classes modulo (q2-1)q. By means of the proof of this result we have determined all possible numerators k for 2≤ q≤ 50, the case q=1 being trivial. The result of this search suggests a conjecture about all possible values k/q, (k,q)=1, of Dedekind sums S(a,b) for an arbitrary q∈\mathbb N.