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

Abstract

This paper provides a survey of particular values of Ramanujan's theta function $\phi (q)={\sum }_{n=-\infty }^{\infty }{q}^{n^2}$, 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.

11 pages, final version, to appear in the second special volume of the Hardy-Ramanujan Journal dedicated to the memory of Srinivasa Ramanujan

References in corpus (1)

• The three missing terms in Ramanujan's septic theta function identity