A generalization of the reciprocity law of multiple Dedekind sums
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, p. 361-377
Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad.In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws.
Plusieurs sommes multiples de Dedekind ont été introduites par B.C.Berndt, L.Carlitz, S.Egami, D.Zagier et A.Bayad. Dans cet article, après avoir remarqué la forme de Jacobi dans Bayad [4], la fonction cotangente dans Zagier [23], le résultat d’Egami sur les fonctions cotangentes [14] et leurs lois de reciprocité, nous étudions un cas spécial de la forme de Jacobi de Bayad [4] et déduisons une généralisation du résultat d’Egami sur les fonctions cotangentes et une généralisation du résultat de Zagier. De plus, nous considérons leurs lois de réciprocité.
DOI : https://doi.org/10.5802/aif.2261
Classification:  11A15,  11B68,  11F20,  11F23,  11F50
Keywords: Dedekind sums, reciprocity law, Jacobi forms
@article{AIF_2007__57_2_361_0,
     author = {Asano, Masahiro},
     title = {A generalization of the reciprocity law of multiple Dedekind sums},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     pages = {361-377},
     doi = {10.5802/aif.2261},
     zbl = {1158.11022},
     mrnumber = {2310944},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_2_361_0}
}
A generalization of the reciprocity law of multiple Dedekind sums. Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 361-377. doi : 10.5802/aif.2261. https://aif.centre-mersenne.org/item/AIF_2007__57_2_361_0/

[1] Atiyah, M. F.; Hirzebruch, F. Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc., Tome 65 (1959), pp. 276-281 | Article | MR 110106 | Zbl 0142.40901

[2] Atiyah, M. F.; Hirzebruch, F. Cohomologie-operationen und charakteristische klassen, Math. Z., Tome 77 (1961), pp. 149-187 | Article | MR 156361 | Zbl 0109.16002

[3] Atiyah, M. F.; Singer, I. M. The index of elliptic operators, Ann. of Math., Tome 87 (1968), pp. 546-604 | Article | MR 236952 | Zbl 0164.24301

[4] Bayad, A. Sommes de Dedekind elliptiques et formes de Jacobi, Ann. Inst. Fourier, Tome 51 (2001) no. 1, pp. 29-42 | Article | Numdam | MR 1821066 | Zbl 1034.11030

[5] Bayad, A.; Robert, G. Amélioration d’une congruence pour certains éléments de Stickelberger quadratiques, Bull. Soc. Math. France, Tome 125 (1997), pp. 249-267 | Numdam | MR 1478032 | Zbl 0895.11021

[6] Bayad, A.; Robert, G. Note sur une forme de Jacobi méromorphe, C.R.A.S., Tome 325 (1997), pp. 455-460 | MR 1692306 | Zbl 0885.11035

[7] Beck, M. Dedekind cotangent sums, Acta Arith., Tome 109 (2003) no. 2, pp. 109-130 | Article | MR 1980640 | Zbl 1061.11043

[8] Berndt, B. C. Reciprocity theorems for Dedekind sums and generalizations, Advances in Math., Tome 23 (1977), pp. 285-316 | Article | MR 429711 | Zbl 0342.10014

[9] Berndt, B. C.; Dieter, U. Sums involving the greatest integer function and Riemann-Stieltjes integration, J. Reine Angew. Math., Tome 337 (1982), pp. 208-220 | MR 676053 | Zbl 0487.10002

[10] Carlitz, L. A note on generalized Dedekind sums, Duke Math. J., Tome 21 (1954), pp. 399-404 | Article | MR 62766 | Zbl 0057.03802

[11] Carlitz, L. A theorem on generalized Dedekind sums, Acta Arith., Tome 11 (1965), pp. 253-260 | MR 182604 | Zbl 0131.28801

[12] Carlitz, L. Many term relations for multiple Dedekind sums, Indian J. Math., Tome 20 (1978), pp. 77-89 | MR 603918 | Zbl 0418.10013

[13] Dieter, U. Pseudo-random numbers : the exact distribution of pairs, Math. of Computation, Tome 25 (1971), pp. 855-883 | MR 298727 | Zbl 0257.65010

[14] Egami, S. An elliptic analogue of multiple Dedekind sums, Compositio Math., Tome 99 (1995), pp. 99-103 | Numdam | MR 1352569 | Zbl 0838.11029

[15] Fukuhara, S.; Yui, N. Elliptic Apostol sums and their reciprocity laws, Trans. Amer. Math. Soc., Tome 356 (2004) no. 10, pp. 4237-4254 | Article | MR 2058844 | Zbl 1055.11028

[16] Harder, G. Periods integrals of cohomology classes which are represented by Eisenstein series, Proc. Bombay Colloquium 1979, Springer Verlag (1981), pp. 41-115 | MR 633658 | Zbl 0497.22021

[17] Hirzebruch, F. Topological methods in algebraic geometry, Springer Verlag, Berlin-Heidelberg-New York (1966) | MR 202713 | Zbl 0138.42001

[18] Hirzebruch, F.; Berger, T.; Jung, R. Manifolds and modular forms, Vieweg Verlag, Aspects of Mathematics, Tome E20 (1992) | MR 1189136 | Zbl 0767.57014

[19] Ito, H. A function on the upper half space which is analogous to imaginary part of logη(z), J. Reine Angew. Math., Tome 373 (1987), pp. 148-165 | Article | MR 870309 | Zbl 0601.10021

[20] Ito, H. On a property of elliptic Dedekind sums, J. Number Th., Tome 27 (1987), pp. 17-21 | Article | MR 904003 | Zbl 0624.10018

[21] Rademacher, H. Generalization of the reciprocity formula for Dedekind sums, Duke Math. J., Tome 21 (1954), pp. 391-397 | Article | MR 62765 | Zbl 0057.03801

[22] Sczech, R. Dedekindsummen mit elliptischen funktionen, Invent. Math., Tome 76 (1984), pp. 523-551 | Article | MR 746541 | Zbl 0521.10021

[23] Zagier, D. Higher order Dedekind sums, Math. Ann., Tome 202 (1973), pp. 149-172 | Article | MR 357333 | Zbl 0237.10025