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arXiv · 2609.35893

Proofs of some conjectures by Bringmann, Han, Heim, and Kane on periodic sign changes in eta-quotients

Jing Jin·Olivia X. M. Yao·Shiwen Zhou
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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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Jing Jin, Olivia X. M. Yao, Shiwen Zhou. 2026-09-27. Proofs of some conjectures by Bringmann, Han, Heim, and Kane on periodic sign changes in eta-quotients. https://arxiv.org/abs/2609.35893

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