English

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

Number Theory 2022-12-23 v2 Classical Analysis and ODEs History and Overview

Abstract

This paper provides a survey of particular values of Ramanujan's theta function φ(q)=n=qn2\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}, when q=eπnq=e^{-\pi\sqrt{n}}, where nn 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.

Keywords

Cite

@article{arxiv.2112.11882,
  title  = {Explicit values for Ramanujan's theta function $\varphi(q)$},
  author = {Bruce C. Berndt and Örs Rebák},
  journal= {arXiv preprint arXiv:2112.11882},
  year   = {2022}
}

Comments

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

R2 v1 2026-06-24T08:27:52.909Z