A Proof of the Third Borwein Conjecture

Abstract

We study the coefficients of $S_n(q)=\prod_{j=1}^n\prod_{s=1}^4(1-q^{5j-s})$. An effective four-peak analysis gives the sign pattern predicted by the third Borwein conjecture for every $n\ge1750$ and every coefficient. The essential cancellation in residue classes $3$ and $4$ is retained as an exact factor $e^{-5z}$ in the combined amplitude, leading to the shifted saddle point equation $d-5n=n^2β(t)$. A two-layer partition injection provides the linear boundary needed to join this analysis to small degrees. All continuous parameter estimates have explicit constants; their finite arithmetic comparisons are supplied as rational certificates. We also describe exact integer verification. Combining the analytic theorem with the author's reported completion of the finite verification for $1\le n\le1749$ gives the conjectured sign pattern for every positive integer $n$. The available supplementary coefficient record covers $1\le n\le500$; the reported full-range computation is identified separately.