On Dedekind sums with equal values

2014/04/17 by Kurt Girstmair, Girstmair, Kurt
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1404.4428

8 pages

arxiv created 2014/04/17 · arxiv updated 2014/04/18

Abstract

Dedekind sums s(m,n) occur in many fields of mathematics. Since s(m₁,n)=s(m₂,n) if m₁≡ m₂ mod n, it is natural to ask which of the Dedekind sums s(m,n), 0≤ m<n, take equal values. So far no simple criterion is known by which the equality of s(m₁,n) and s(m₂,n) could be decided. In this note we show how to obtain non-obvious examples of equal Dedekind sums. We consider two cases which mark the extreme possibilities for the argument n, namely, n a prime power and n square-free. Whereas we can give a partial overview of equal Dedekind sums in the prime power case, such an overview seems to be much more difficult to obtain in the square-free case.

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