## 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)].

### MSC:

 11F20 Dedekind eta function, Dedekind sums 11B68 Bernoulli and Euler numbers and polynomials

### Keywords:

multiple Dedekind sum; reciprocity formula

### Citations:

Zbl 0039.03801; Zbl 0632.10023; Zbl 0057.03802
Full Text:

### References:

 [1] T. M. Apostol, Generalized Dedekind sums and transformation formula of certain Lambert series, Duke Math. J. 17 (1950), no.2, 147-157. · Zbl 0039.03801 [2] L. Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953), no.3, 513-522. · Zbl 0057.03701 [3] L. Carlitz, The reciprocity theorem for Dedekind sums, Pacific J. Math. 3 (1953), no.3, 523-527. · Zbl 0057.03703 [4] L. Carlitz, A note on generalized Dedekind sums, Duke Math. J. 21 (1954), no.3, 399-403. · Zbl 0057.03802 [5] L. Carlitz, Many-term relations for multiple Dedekind sums, Indian J. Math. 20 (1978), no.1, 77-89. · Zbl 0418.10013 [6] K. Kozuka, Dedekind type sums attached to Dirichlet characters, Kyusyu J. Math. 58 (2004), no.1, 1-24. · Zbl 1060.11024 [7] A. Kudo, On $$p$$-adic Dedekind sums (II), Mem. Fac. Sci. Kyushu Univ. 45 (1991), no.2, 245-284. · Zbl 0751.11031 [8] C. Nagasaka, On generalized Dedekind sums attached to Dirichlet characters, J. Number Theory 19 (1984), no.3, 374-383. · Zbl 0551.10022 [9] N. E. Nörlund, Vorlesungen über Differenzenrechnung , Springer, Berlin, 1924. · JFM 50.0318.04 [10] C. Snyder, $$p$$-adic interpolation of Dedekind sums, Bull. Austral. Math. Soc. 38 (1988), no.2, 293-301. · Zbl 0632.10023 [11] H. Tsumura, On a $$p$$-adic interpolation of the generalized Euler numbers and its applications, Tokyo J. Math. 10 (1987), no.2, 281-293. · Zbl 0641.12007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.