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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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.


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Additional Information

Shinji Fukuhara
Affiliation: Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187, Japan
Email: fukuhara@tsuda.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05291-0
PII: S 0002-9939(99)05291-0
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