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arXiv · 1411.4092

On the Inversion Polynomial for Dedekind Sums

Yiwang Chen·Nicholas Dunn·Campbell Hewett·Shashwat Silas
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Abstract

We introduce the inversion polynomial for Dedekind sums $f_b(x)=\sum x^{\operatorname{inv}(a,b)}$ to study the number of $s(a,b)$ which have the same value for 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 restriction on equal Dedekind sums based on the reciprocity formula.

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Yiwang Chen, Nicholas Dunn, Campbell Hewett, Shashwat Silas. 2014-11-15. On the Inversion Polynomial for Dedekind Sums. https://doi.org/10.1142/s1793042115501067

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