Ramanujan's remarkable product of theta-functions

Abstract

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.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1017/s0013091500024032

Publication Date

1-1-1997

Journal Title

Proceedings of the Edinburgh Mathematical Society

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