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

Full-text PDF Free Access

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.

**1.**T. M. Apostol,*Generalized Dedekind sums and transformation formulae of certain Lambert series*, Duke Math. J.**17**(1950), 147–157. MR**0034781****2.**Fukuhara, S.: Modular forms, generalized Dedekind symbols and period polynomials. Math. Ann., 83-101 (1998) CMP**98:07****3.**F. Hirzebruch,*The signature theorem: reminiscences and recreation*, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Princeton Univ. Press, Princeton, N.J., 1971, pp. 3–31. Ann. of Math. Studies, No. 70. MR**0368023****4.**Friedrich Hirzebruch, Thomas Berger, and Rainer Jung,*Manifolds and modular forms*, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR**1189136****5.**F. Hirzebruch and D. Zagier,*The Atiyah-Singer theorem and elementary number theory*, Publish or Perish, Inc., Boston, Mass., 1974. Mathematics Lecture Series, No. 3. MR**0650832****6.**Don Zagier,*Higher dimensional Dedekind sums*, Math. Ann.**202**(1973), 149–172. MR**0357333****7.**Don Zagier,*Periods of modular forms and Jacobi theta functions*, Invent. Math.**104**(1991), no. 3, 449–465. MR**1106744**, 10.1007/BF01245085

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11F20,
11F67,
11F11,
11M35,
33E20

Retrieve articles in all journals with MSC (1991): 11F20, 11F67, 11F11, 11M35, 33E20

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

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