Bruce C Berndt ; Örs Rebák

Explicit Values for Ramanujan's Theta Function ϕ(q)
hrj:8923 
HardyRamanujan 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
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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.