A Proof of the Third and Cubic Borwein Conjectures

Abstract

We establish the coefficient sign patterns in the Third Borwein conjecture and the modulus-three Cubic Borwein conjecture. The analytic arguments apply for $n\ge 1750$ and $n\ge 500$, respectively. They combine exact dissections and positive coefficient identities near the boundary with saddle point estimates that preserve cancellation between primitive-root contributions. Paired Gaussian estimates remove the leading odd error, and explicit remainder bounds cover the complementary contours. The remaining finite intervals are checked by exact integer arithmetic. The Third finite verification through $1749$ is author-confirmed; the Cubic verification through $500$ is supported by two complete integer implementations and additional coefficient crosschecks.