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Generalized Dedekind symbols associated with the Eisenstein series
Author(s):
Shinji
Fukuhara
Journal:
Proc. Amer. Math. Soc.
127
(1999),
2561-2568.
MSC (1991):
Primary 11F20, 11F67;
Secondary 11F11, 11M35, 33E20
Posted:
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.
References:
- 1.
- Apostol, T. M.: Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J., 147-157 (1950) MR 11:641g
- 2.
- Fukuhara, S.: Modular forms, generalized Dedekind symbols and period polynomials. Math. Ann., 83-101 (1998) CMP 98:07
- 3.
- Hirzebruch, F.: The signature theorem: reminiscences and recreation. In: Prospects in Mathematics (Ann. Math. Studies No. 70), pp. 3-31. Princeton 1971 MR 51:4265
- 4.
- Hirzebruch, F., Berger, T., Jung, R.: Manifolds and Modular forms. Bonn: Vieweg 1992 MR 94d:57001
- 5.
- Hirzebruch, F., Zagier, D.: The Atiyah-Singer theorem and elementary number theory. Berkeley: Publish or Perish 1974 MR 58:31291
- 6.
- Zagier, D.: Higher dimensional Dedekind sums. Math. Ann., 149-172 (1973) MR 50:9801
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- Zagier, D.: Periods of modular forms and Jacobi theta functions. Invent. Math., 449-465 (1991) MR 92e:11052
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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:
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
Posted:
May 4, 1999
Additional Notes:
The author wishes to thank Professor N. Yui for her useful advice
Communicated by:
Dennis A. Hejhal
Copyright of article:
Copyright
1999,
American Mathematical Society
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