Multiple Dedekind sums attached to Dirichlet characters. (English) Zbl 1197.11053

For a nonnegative integer \(m\), let \(B_m(x)\) denote the \(m\)th Bernoulli function. Then the classical Dedekind sum \(s(h,k)\) with \(h,k\in\mathbb Z\) and \(k>0\) is defined by \[ s(h,k)=\sum_{a\mod k}\left(\left(\frac ak\right)\right) \left(\left(\frac{ah}k\right)\right), \] where \(((x))=B_1(x)\) if \(x\not\in\mathbb Z\) and \(((x))=0\) if \(x\in\mathbb Z\). The original reciprocity formula for Dedekind sums states that \[ 12\,hk\left(s(h,k)+s(k,h)\right)=h^2-3hk+k^2+1 \] for \(h,k>0\) with \((h,k)=1\). Generalizations of this result have been obtained by Apostol, Carlitz, and many others.
In the paper under review, the author first introduces the notion of multiple Dedekind sums attached to Dirichlet characters and then proves formulas that contain the original reciprocity formula for Dedekind sums and its generalizations due to T. M. Apostol [Duke Math. J. 17, 147–157 (1950; Zbl 0039.03801)], C. Snyder [Bull. Aust. Math. Soc. 37, 293–302 (1988; Zbl 0632.10023)], and L. Carlitz [Duke Math. J. 21, 399–403 (1954; Zbl 0057.03802)].


11F20 Dedekind eta function, Dedekind sums
11B68 Bernoulli and Euler numbers and polynomials
Full Text: DOI Euclid


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