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On reciprocity formulae for inhomogeneous and homogeneous Dedekind sums

Published online by Cambridge University Press:  24 October 2008

R. R. Hall
Affiliation:
Department of Mathematics, University of York, York Y01 5DD
J. C. Wilson
Affiliation:
Department of Mathematics, University of York, York Y01 5DD

Extract

A number of authors, including Apostol [1], Carlitz [2], Mikolás [5] and Rademacher [9] have obtained linear relations for the Dedekind sums

(the inhomogeneous sum) and the homogeneous sum

Here denotes the periodic extension into ℝ of the Bernoulli polynomial Bm(X) on [0, 1] given by the relation

with the exception that we define

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Apostol, T. M.. Generalized Dedekind sums and the transformation formulae of certain Lambert series. Duke Math. J. 17 (1950), 147157.CrossRefGoogle Scholar
[2]Carlitz, L.. Some theorems on generalized Dedekind sums. Pacific J.Math. 3 (1953), 513522.CrossRefGoogle Scholar
[3]Greaves, G. R. H., Hall, R. R., Huxley, M. N. and Wilson, J. C.. Multiple Franel integrals. Mathematika, 40 (1993), 5069.CrossRefGoogle Scholar
[4]Hall, R. R.. The distribution of squarefree numbers. J. reine angew Math. 394 (1989), 107117.Google Scholar
[5]Mikolas, M.. On certain sums generating the Dedekind sums and their reciprocity laws. Pacific J. Math. 7 (1957), 11671178.CrossRefGoogle Scholar
[6]Montgomery, H. L. and Vaughan, R. C.. A basic inequality. In Gongreso de teoria de los numeros (Zarautz conf. proc.), ed. Aparicio, E., Calderon, C. and Peral, J. C. (1984).Google Scholar
[7]Montgomery, H. L. and Vaughan, R. C.. On the distribution of reduced residues. Annals of Math. 123 (1986), 311333.CrossRefGoogle Scholar
[8]Rademacher, H.. Generalization of the reciprocity formula for Dedekind sums. Duke Math. J. 21 (1954), 391397.CrossRefGoogle Scholar
[9]Rademacher, H. and Grosswald, E.. Dedekind Sums (Carus Math. Monographs, 16).Google Scholar