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Franel integrals of order four

Part of: Discontinuous groups and automorphic forms Real functions

Published online by Cambridge University Press:  09 April 2009

Richard J. McIntosh
Richard J. McIntosh
Affiliation:
Department of Mathematics and StatisticsUniversity of ReginaRegina, SaskatchewanCanadaS4S 0A2 e-mail: mcintosh@ninja.math.uregina.ca
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Abstract

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Let ((x)) = x −⌊x⌋−1/2 be the swatooth function. If a, b, c and e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

MSC classification

Secondary: 26A09: Elementary functions 11F20: Dedekind eta function, Dedekind sums

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 2 , April 1996 , pp. 192 - 203
Copyright
Copyright © Australian Mathematical Society 1996

References

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