10.46298/hrj.2022.8923
https://hrj.episciences.org/8923
Berndt, Bruce C
Bruce C
Berndt
Rebák, Örs
Örs
Rebák
Explicit Values for Ramanujan's Theta Function ϕ(q)
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.
episciences.org
theta functions
explicit values
modular equations
Ramanujan's lost notebook
2010 Mathematics Subject Classification. 11-02; 11F27
[MATH]Mathematics [math]
2022-01-09
2022-01-09
2022-01-09
en
journal article
https://hal.archives-ouvertes.fr/hal-03498537v1
2804-7370
https://hrj.episciences.org/8923/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021
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