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A generating function of higher-dimensional Apostol–Zagier sums and its reciprocity law

✍ Scribed by Shinji Fukuhara; Noriko Yui

Publisher: Elsevier Science
Year: 2006
Tongue: English
Weight: 199 KB
Volume: 117
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2005.05.008

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✦ Synopsis

We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z → ∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol-Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.

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