episciences.org_8923_1669886071
1669886071
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
09
2022
Volume 44  Special...
Explicit Values for Ramanujan's Theta Function ϕ(q)
Bruce C
Berndt
Örs
Rebák
https://orcid.org/0000000233420664
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.
01
09
2022
8923
https://hal.archivesouvertes.fr/hal03498537v1
10.46298/hrj.2022.8923
https://hrj.episciences.org/8923

https://hrj.episciences.org/8923/pdf