Explicit Values for Ramanujan's Theta Function ϕ(q)

Bruce C Berndt, Örs Rebák
2022 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
more » ... d in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.
doi:10.46298/hrj.2022.8923 fatcat:iyi3ctpddffwro4dv2vr7icjzq