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arXiv · 1812.09482

On a recent reciprocity formula for Dedekind sums

Kurt Girstmair
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Abstract

Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Recently, Du and Zhang proved the following reciprocity formula. If $a$ and $b$ are odd natural numbers, $(a,b)=1$, then $$ S(2a^*,b)+S(2b^*,a)=\frac{a^2+b^2+4}{2ab}-3, $$ where $aa^*\equiv 1\mod b$ and $bb^* \equiv 1 \mod a$. In this paper we show that this formula is a special case of a series of similar reciprocity formulas. Whereas Du and Zhang worked with the connection of Dedekind sums and values of $L$-series, our main tool is the three-term relation for Dedekind sums.

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Kurt Girstmair. 2018-12-22. On a recent reciprocity formula for Dedekind sums. https://arxiv.org/abs/1812.09482

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