Hardy-Ramanujan Journal |

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.

Source: HAL:hal-03498537v1

Volume: Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021

Published on: January 9, 2022

Accepted on: January 9, 2022

Submitted on: January 6, 2022

Keywords: theta functions,explicit values,modular equations,Ramanujan's lost notebook,2010 Mathematics Subject Classification. 11-02; 11F27,[MATH]Mathematics [math]

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