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Generalized Dedekind symbols associated
with the Eisenstein series

Author: Shinji Fukuhara
Journal: Proc. Amer. Math. Soc. 127 (1999), 2561-2568
MSC (1991): Primary 11F20, 11F67; Secondary 11F11, 11M35, 33E20
Published electronically: May 4, 1999
MathSciNet review: 1657743
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Abstract | References | Similar Articles | Additional Information

Abstract: We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.

References [Enhancements On Off] (What's this?)

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Additional Information

Shinji Fukuhara
Affiliation: Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187, Japan

Keywords: Dedekind sum, Eisenstein series, polylogarithm
Received by editor(s): October 7, 1997
Published electronically: May 4, 1999
Additional Notes: The author wishes to thank Professor N. Yui for her useful advice
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1999 American Mathematical Society