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Proofs of some conjectures by Bringmann, Han, Heim, and Kane on periodic sign changes in eta-quotients
Abstract
Bringmann, Han, Heim, and Kane recently investigated periodic sign patterns in the Fourier coefficients of weakly holomorphic modular forms arising from eta-quotients and posed numerous conjectures in the appendix to their work. Using classical theta-function identities, dissections, and coefficientwise positivity, we prove 72 of these conjectural sign patterns. The eta-quotients considered here have periods 2, 3, 4, 5, 6, 8, and 12. Our proofs are elementary at the level of q-series and yield explicit generating functions for the coefficients in each relevant residue class.
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BibTeX
@misc{jin2026proofs,
title = {{Proofs of some conjectures by Bringmann, Han, Heim, and Kane on periodic sign changes in eta-quotients}},
author = {Jing Jin and Olivia X. M. Yao and Shiwen Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35893},
eprint = {2609.35893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35893},
}
RIS
TY - UNPB
TI - Proofs of some conjectures by Bringmann, Han, Heim, and Kane on periodic sign changes in eta-quotients
AU - Jing Jin
AU - Olivia X. M. Yao
AU - Shiwen Zhou
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.35893
UR - https://arxiv.org/abs/2609.35893
AB - Bringmann, Han, Heim, and Kane recently investigated periodic sign patterns in the Fourier coefficients of weakly holomorphic modular forms arising from eta-quotients and posed numerous conjectures in the appendix to their work. Using classical theta-function identities, dissections, and coefficientwise positivity, we prove 72 of these conjectural sign patterns. The eta-quotients considered here have periods 2, 3, 4, 5, 6, 8, and 12. Our proofs are elementary at the level of q-series and yield explicit generating functions for the coefficients in each relevant residue class.
ER -