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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ramanujan's class invariants,
Kronecker's limit formula,
and modular equations


Authors: Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang
Journal: Trans. Amer. Math. Soc. 349 (1997), 2125-2173
MSC (1991): Primary 11R29; Secondary 11R04, 11R37, 11R42, 11F27, 33D10
MathSciNet review: 1376539
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Abstract | References | Similar Articles | Additional Information

Abstract: In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker's limit formula, the second employs modular equations, and the third uses class field theory to make Watson's ``empirical method''rigorous.


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Additional Information

Bruce C. Berndt
Affiliation: Department of Mathematics, 1409 West Green Street, University of Illinois, Urbana, Illinois 61801
Email: berndt@math.uiuc.edu

Heng Huat Chan
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, National Chung Cheng University, Minhsiung, Chiyai 621, Taiwan, R.O.C.
Email: hhchan@mthmp.math.ccu.edu.tw

Liang-Cheng Zhang
Affiliation: Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804
Email: liz917f@cnas.smsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01738-8
PII: S 0002-9947(97)01738-8
Received by editor(s): March 16, 1995
Received by editor(s) in revised form: November 14, 1995
Dedicated: In Memory of Jerry Keiper
Article copyright: © Copyright 1997 American Mathematical Society