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Reciprocity theorems for Dedekind sums and generalizations. (English) Zbl 0342.10014
This work is concerned with various types of Dedekind sums involving the first Bernoulli function only, i.e., \(((x))\) in the customary notation. All of the Dedekind sums examined here occur in the transformation formulae of functions akin to the logarithm of the Dedekind eta-function. Reciprocity and three-term relations are established for the sums. The methods are analytic, but transformation formulae are not used. Most of the theorems are not new. However, in some cases, the only previously known proofs utilized transformation formulas. Furthermore, the proofs given here are frequently simpler and shorter than other proofs.
Reviewer: Bruce C. Berndt

MSC:
11F20 Dedekind eta function, Dedekind sums
11B68 Bernoulli and Euler numbers and polynomials
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[1] Ayoub, R., An introduction to the analytic theory of numbers, (1963), American Mathematical Society Chicago, Ill. · Zbl 0128.04303
[2] Berndt, B.C., Generalized Dedekind eta-functions and generalized Dedekind sums, Trans. amer. math. soc., 178, 495-508, (1973) · Zbl 0262.10015
[3] Berndt, B.C., Character transformation formulae similar to those for the Dedekind eta-function, (), 9-30, No. 24
[4] Berndt, B.C., Generalized Eisenstein series and modified Dedekind sums, J. reine angew. math., 272, 182-193, (1975) · Zbl 0294.10018
[5] Berndt, B.C., A new proof of the reciprocity theorem for Dedekind sums, Elem. math., 29, 93-94, (1974) · Zbl 0283.10010
[6] Berndt, B.C., On Eisenstein series with characters and the values of Dirichlet L-functions, Acta arith., 28, 299-320, (1975) · Zbl 0279.10023
[7] Berndt, B.C., Dedekind sums and a paper of G. H. Hardy, J. London math. soc., 13, 2, 129-137, (1976) · Zbl 0319.10006
[8] Berndt, B.C.; Schoenfeld, L., Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory, Acta arith., 28, 23-68, (1975) · Zbl 0268.10008
[9] Carlitz, L., A note on generalized Dedekind sums, Duke math. J., 21, 399-403, (1954) · Zbl 0057.03802
[10] Carlitz, L., A further note on Dedekind sums, Duke math. J., 23, 219-223, (1956) · Zbl 0074.03504
[11] Carlitz, L., Generalized Dedekind sums, Math. Z., 85, 83-90, (1964) · Zbl 0122.05104
[12] Carlitz, L., A theorem on generalized Dedekind sums, Acta arith., 11, 253-260, (1965) · Zbl 0131.28801
[13] Carlitz, L., Linear relations among generalized Dedekind sums, J. reine angew. math., 220, 154-162, (1965) · Zbl 0148.27305
[14] Carlitz, L., A three-term relation for the Dedekind-Rademacher sums, Publ. math. debrecen, 14, 119-124, (1967) · Zbl 0167.31403
[15] Dieter, U., Beziehungen zwischen dedekindschen summen, Abh. math. sem. univ. Hamburg, 21, 109-125, (1957) · Zbl 0078.07002
[16] Dieter, U., Das verhalten der kleinschen funktionen log σ_{g, h}(w1, w2) gegenüber modultransformationen und verallgemeinerte dedekindsche summen, J. reine angew. math., 201, 37-70, (1959) · Zbl 0085.02604
[17] Grosswald, E., Dedekind-Rademacher sums, Amer. math. monthly, 78, 639-644, (1971) · Zbl 0212.07701
[18] Grosswald, E., Dedekind-Rademacher sums and their reciprocity formula, J. reine angew. math., 251, 161-173, (1971) · Zbl 0223.10012
[19] Hardy, G.H., On certain series of discontinuous functions connected with the modular functions, Quart. J. math., 36, 93-123, (1905) · JFM 35.0468.03
[20] Iseki, K., A proof of a transformation formula in the theory of partitions, J. math. soc. Japan, 4, 14-26, (1952) · Zbl 0049.31104
[21] Knopp, K., Theory and application of infinite series, (1951), Blackie & Son Providence, R. I. · JFM 54.0222.09
[22] Meyer, C., Über einige anwendungen dedekindscher summen, J. reine angew. math., 198, 143-203, (1957) · Zbl 0079.10303
[23] Meyer, C., Bemerkungen zu den allgemeinen dedekindschen summen, J. reine angew. math., 205, 186-196, (1961) · Zbl 0097.26401
[24] Rademacher, H., Egy reciprocitásképletröl a modulfüggevények elméletéböl, Mat. fiz. lapok, 40, 24-34, (1933)
[25] Rademacher, H., Die reziprozitätsformel für dedekindsche summen, Acta sci. math. (Szeged), 12(B), 57-60, (1950) · Zbl 0037.31104
[26] Rademacher, H., Generalization of the reciprocity formula for Dedekind sums, Duke math. J., 21, 391-397, (1954) · Zbl 0057.03801
[27] Rademacher, H., Some remarks on certain generalized Dedekind sums, Acta arith, 9, 97-105, (1964) · Zbl 0128.27101
[28] Rademacher, H.; Grosswald, E., Dedekind sums, () · Zbl 0251.10020
[29] Zacier, D., Higher dimensional Dedekind sums, Math. ann., 202, 149-172, (1973)
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