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Ramanujan's remarkable product of theta-functions

Published online by Cambridge University Press:  20 January 2009

Bruce C. Berndt ,
Heng Huat Chan  and
Liang-Cheng Zhang
Bruce C. Berndt
Affiliation:
Department of Mathematics, 1409 West Green Street, University of Illinois, Urbana, IL 61801, U.S.A.
Heng Huat Chan
Affiliation:
Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, U.S.A.
Liang-Cheng Zhang
Affiliation:
Department of Mathematics, National Chung Cheng University, Minhsiung, Chiayi 621, Taiwan
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Abstract

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On pages 338 and 339 in his first notebook, Ramanujan records eighteen values for a certain product of theta-functions depending on two integral parameters m and n. When (m, n) = 1, it can be seen that each of these values is a unit. The purpose of this paper is to establish each of these eighteen values and to prove that under certain general conditions this product is indeed a unit. Lastly, we prove that certain quotients of theta-functions are algebraic integers.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 3 , October 1997 , pp. 583 - 612
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

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