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≥1750 and n≥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.
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BibTeX
@misc{ma2026proof,
title = {{A Proof of the Third and Cubic Borwein Conjectures}},
author = {Yicen Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01156},
eprint = {2610.01156},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01156},
}
RIS
TY - UNPB
TI - A Proof of the Third and Cubic Borwein Conjectures
AU - Yicen Ma
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.01156
UR - https://arxiv.org/abs/2610.01156
AB - We establish the coefficient sign patterns in the Third Borwein conjecture and the modulus-three Cubic Borwein conjecture. The analytic arguments apply for n≥1750 and n≥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.
ER -