no. 1
A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity
Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 93-101.

Une loi de réciprocité est établie pour des sommes apparaissant dans les formules de transformations pour les logarithmes des fonctions theta, sommes qui sont les analogues des sommes de Dedekind dans la transformation du logarithme de la fonction eta.

In the transformation formulas for the logarithms of the classical theta-functions, certain sums arise that are analogous to the Dedekind sums in the transformation of the logarithm of the eta-function. A new reciprocity law is established for one of these analogous sums and then applied to prove the law of quadratic reciprocity.

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     title = {A reciprocity congruence for an analogue of the {Dedekind} sum and quadratic reciprocity},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {93--101},
     publisher = {Universit\'e Bordeaux I},
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     number = {1},
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Meyer, Jeffrey L. A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity. Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 93-101. http://archive.numdam.org/item/JTNB_2000__12_1_93_0/

[1] B.C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 303/304 (1978), 332-365. | EuDML | MR | Zbl

[2] L.A. Goldberg, Transformations of theta-functions and analogues of dedekind sums. Ph.D. thesis, Thesis, University of Illinois, Urbana, 1981.

[3] H. Rademacher E. Grosswald, Dedekind sums. Carus Math. Monogr., vol. 16, Mathematical Association of America, Washington, D.C., 1972. | MR | Zbl

[4] H. Montgomery I. Niven, H. Zuckerman, An introduction to the theory of numbers. 5th ed., John Wiley and Sons, New York, 1991. | MR | Zbl

[5] J. Lewittes, Analytic continuation of Eisenstein series. Trans. Amer. Math. Soc. 171 (1972), 469-490. | MR | Zbl

[6] J.L. Meyer, Analogues of dedekind sums. Ph.D. thesis, University of Illinois, Urbana, 1997.

[7] _Properties of certain integer-valued analogues of Dedekind sums. Acta Arithmetica 82 (1997), 229-242. | EuDML | MR | Zbl

[8] H. Rademacher, Topics in analytic number theory. Springer-Verlag, New York, 1973. | MR | Zbl