Reciprocity formulae for general Dedekind-Rademacher sums.

Hall, R.R.; Wilson, J.C.; Zagier, D.

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Reciprocity formulae for general Dedekind-Rademacher sums.

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Wilson, J.C.
Max Planck Society;

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Zagier, D.
Max Planck Institute for Mathematics, Max Planck Society;

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Hall, R., Wilson, J., & Zagier, D. (1995). Reciprocity formulae for general Dedekind-Rademacher sums. Acta Arithmetica, 73(4), 389-396.

Cite as: https://hdl.handle.net/21.11116/0000-0004-38FB-3

Abstract

The general Dedekind-Rademacher sums are defined, for positive integers a, b, c and real numbers x, y, z modulo 1, by the formula Sm, n\pmatrix a amp; b amp; c\\ x amp; y amp; z\endpmatrix= \sumh\pmod c \overline Bm \Biggl( ah+ z\over c- x\Biggr) \overline Bn\Biggl( b h+ z\over c- y\Biggr), where \overline Bm is 1-periodic and equal to the Bernoulli polynomial Bm on the unit interval. The authors prove a reciprocity theorem which is stated in terms of a generating function involving Sm, n for all pairs of non-negative integers m, n. The reciprocity theorem embraces the classical relations (due to Dedekind and Rademacher) and their generalizations (due to Apostol, Berndt, Carlitz and Mikolás) which are known up to now.