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An elliptic analogue of the multiple Dedekind sums. (English) Zbl 0838.11029
D. Zagier [Math. Ann. 202, 149-172 (1973; Zbl 0246.10023)] introduced higher order Dedekind sums in terms of multiple cotangent sums. This paper uses an elliptic function in place of the cotangent function and obtains a corresponding reciprocity law. Some of Zagier’s results are deduced as limiting cases of the elliptic analogue.

MSC:
11F20 Dedekind eta function, Dedekind sums
33E05 Elliptic functions and integrals
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References:
[1] Berndt, B.C. : Reciprocity theorems of Dedekind sums and generalization , Advance in Math. 23 (1977), 285-316. · Zbl 0342.10014
[2] Carlitz, L. : A note on generalized Dedekind sums , Duke Math. J. 21 (1954), 399-403. · Zbl 0057.03802
[3] Carlitz, L. : Many term relations for multiple Dedekind sums . Indian J. Math. 20 (1978), 77-89. · Zbl 0418.10013
[4] Eichler, M. and Zagier, D. : The Theory of Jacobi Forms , Progress in Math. 55, Birkhäuser, 1985. · Zbl 0554.10018
[5] Hirzebruch, F. , Berger, T. and Jung, R. : Manifolds and Modular forms , Aspects of Math. E. 20, Vieweg, 1992. · Zbl 0767.57014
[6] Hirzebruch, F. and Zagier, D. : The Atiyah-Singer Theorem and Elementary Number Theory , Math. Lecture Series 3, Publish or Perish Inc., 1974. · Zbl 0288.10001
[7] Ito, H. : A function on the upper half space which is analogous to imaginary part of log \eta (z), J. reine angew. Math. 373 (1987), 148-165. · Zbl 0601.10021
[8] Ito, H. : On a property of elliptic Dedekind sums , J. Number Th. 27 (1987), 17-21. · Zbl 0624.10018
[9] Sczech, R. : Dedekindsummen mit elliptischen Functionen , Invent. Math. 76 (1984), 523-551. · Zbl 0521.10021
[10] Zagier, D. : Higher order Dedekind sums , Math. Ann. 202 (1973), 149-172. · Zbl 0237.10025
[11] Zagier, D. : Equivariant Pontrjagin Classes and Application to Orbit Spaces , Lecture Note in Math. 290, Springer, 1972. · Zbl 0238.57013
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