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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Elliptic Apostol sums and their reciprocity laws
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by Shinji Fukuhara and Noriko Yui PDF
Trans. Amer. Math. Soc. 356 (2004), 4237-4254 Request permission

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
  • MR Author ID: 186000
  • Email: yui@mast.queensu.ca
  • 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.
  • © Copyright 2004 American Mathematical Society
  • 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
  • MathSciNet review: 2058844