Explaining Ramanujan's "Unclosed" Comment in His Last Letter to Hardy

Daniel Parry

Explaining Ramanujan's "Unclosed" Comment in His Last Letter to Hardy

Number Theory 2016-01-05 v3

Authors: Daniel Parry

Abstract

We study the $q$ -series $F(q) = \sum q^{m(m+1)/2}/(q)_m^2$ and offer some commentary on why Ramanujan may have thought $F(q)$ does not "close."

Keywords

q-series Ramanujan Hardy last letter unclosed comment q-Pochhammer symbol analytic combinatorics

Cite

@article{arxiv.1512.07436,
  title  = {Explaining Ramanujan's "Unclosed" Comment in His Last Letter to Hardy},
  author = {Daniel Parry},
  journal= {arXiv preprint arXiv:1512.07436},
  year   = {2016}
}

Comments

Typo Log V2: fixed typo of summation index n->j on page 12 V3: Fixed minus sign error on $m$ in Remark 1. V3: Orphan n->j change on Equation 3.7 fixed. V3: Deleted extra ) in equation (4.2). V3: Fixed mistake with the variable transformation $\sqrt[4]{5}\to 1/\sqrt{4]{5}$ throughout the manuscript V3: Added more context to the dot product computation

Explaining Ramanujan's "Unclosed" Comment in His Last Letter to Hardy

Abstract

Keywords

Cite

Comments

Related papers