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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-03498537
Contributor : Srinivas Kotyada Connect in order to contact the contributor
Submitted on : Tuesday, December 21, 2021 - 9:38:44 AM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM

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Bruce C Berndt, Örs Rebák. Explicit Values for Ramanujan's Theta Function ϕ(q). Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2022, Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, pp.41 -- 50. ⟨10.46298/hrj.2022.8923⟩. ⟨hal-03498537⟩

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