Dedekind sums s(a, b) and inversions modulo b

作者：Chen Yiwang^*; Dunn Nicholas; Hewett Campbell; Silas Shashwat

来源：International Journal of Number Theory, 2015, 11(8): 2325-2339.

DOI：10.1142/S1793042115501067

• Dedekind sum
• Dedekind sum reciprocity law
• inversion number
• inversion polynomial

We introduce the inversion polynomial for Dedekind sums f(b)(x) = Sigma x(inv(a,b)) to study the number of s(a, b) which have the same value for a 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 sufficient condition for the inequality of two Dedekind sums based on the reciprocity formula.

• 出版日期2015-12