Explicit values for Ramanujan's theta function $\varphi(q)$

TL;DR

This paper surveys specific values of Ramanujan's theta function at special points, detailing methods for their evaluation, classical results, and recent completions of previously incomplete evaluations from Ramanujan's lost notebook.

Contribution

It provides a comprehensive overview of known and newly completed evaluations of Ramanujan's theta function at special rational points, including methods and proofs.

Findings

Classical values of theta function are summarized.

Newly completed evaluation from Ramanujan's lost notebook is presented.

Methods for evaluating theta function values are described.

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.