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Elliptic Apostol sums and their reciprocity laws


Authors: Shinji Fukuhara and Noriko Yui
Journal: Trans. Amer. Math. Soc. 356 (2004), 4237-4254
MSC (2000): Primary 11F20; Secondary 33E05, 11F11
DOI: https://doi.org/10.1090/S0002-9947-04-03481-6
Published electronically: May 10, 2004
MathSciNet review: 2058844
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Abstract: We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter $\tau$ having positive imaginary part. When $\tau\to i\infty$, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable $\tau$. We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).


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Additional Information

Shinji Fukuhara
Affiliation: Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187-8577, Japan
Email: fukuhara@tsuda.ac.jp

Noriko Yui
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email: yui@mast.queensu.ca

DOI: https://doi.org/10.1090/S0002-9947-04-03481-6
Keywords: Generalized Dedekind sums (Apostol sums), elliptic functions, elliptic Apostol sums, modular forms, reciprocity laws
Received by editor(s): September 30, 2002
Received by editor(s) in revised form: August 7, 2003
Published electronically: May 10, 2004
Additional Notes: The first author was partially supported by Grant-in-Aid for Scientific Research (C)12640089, Ministry of Education, Sciences, Sports and Culture, Japan.
The second author was partially supported by a Research Grant from NSERC, Canada.
Article copyright: © Copyright 2004 American Mathematical Society

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