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

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."

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