Bruce C Berndt ; Örs Rebák - Explicit Values for Ramanujan's Theta Function ϕ(q)

hrj:8923 - Hardy-Ramanujan Journal, January 9, 2022, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021 - https://doi.org/10.46298/hrj.2022.8923
Explicit Values for Ramanujan's Theta Function ϕ(q)

Authors: Bruce C Berndt ; Örs Rebák

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.

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]