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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

REFERENCES

1.Berndt, B. C., Ramanujan's Notebooks, Part III (Springer-Verlag, New York, 1991).Google Scholar
2.Berndt, B. C. and Chan, H. H., Ramanujan's explicit values for the classical theta-function, Mathematika 42 (1995), 278–294.CrossRefGoogle Scholar
3.Berndt, B. C., Chan, H. H. and Zhang, L.-C., Ramanujan's class invariants, Kronecker's limit formula, and modular equations, Trans. Amer. Math. Soc., to appear.Google Scholar
4.Berndt, B. C., Chan, H. H. and Zhang, L.-C., Explicit evaluations of the Rogers-Ramanujan continued fraction 480 (1996), 141–159.Google Scholar
5.Borevich, Z. I. and Shafarevich, I. R., Number Theory (Academic Press, New York, 1966).Google Scholar
6.Borwein, J. M. and Borwein, P. B., Pi and the AGM (Wiley, New York, 1987).Google Scholar
7.Buell, D. A., Binary Quadratic Forms, Classical Theory and Modern Computations (Springer-Verlag, New York, 1989).Google Scholar
8.Cohen, H., A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1993).CrossRefGoogle Scholar
9.Cox, D. A., Primes of the Form x2 + ny2 (Wiley, New York, 1989).Google Scholar
10.Kortum, R. and Mcniel, G., A Table of Periodic Continued Fractions (Lockheed Aircraft Corporation, Sunnyvale, CA).Google Scholar
11.Pohst, M. and Zassenhaus, H., Algorithmic Algebraic Number Theory (Cambridge University Press, Cambridge, 1989).CrossRefGoogle Scholar
12.Ramanathan, K. G., Some applications of Kronecker's limit formula, J. Indian Math. Soc. 52 (1987), 71–89.Google Scholar
13.Ramanujan, S., Modular equations and approximations to π, Quart. J. Math. (Oxford) 45 (1914), 350–372.Google Scholar
14.Ramanujan, S., Notebooks (2 volumes) (Tata Institute of Fundamental Research, Bombay, 1957).Google Scholar
15.Ramanujan, S., Collected Papers (Chelsea, New York, 1962).Google Scholar
16Stark, H. M., Values of L-functions at s = 1 I. L-functions for quadratic forms, Adv. Math. 7 (1971), 301–343.CrossRefGoogle Scholar
17.Weber, H., Lehrbuch der Algebra, drifter Band (Chelsea, New York, 1961).Google Scholar