A proof of the Berkovich-Dhar conjecture modulo three
Abstract
Let Pn(q)=∏j=1n(1−q3j−2)(1−q3j−1). For each p∈{4,5,6,7,8}, we prove that the coefficients of Pn(q)p in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coefficients are in fact strictly positive for 5≤ p≤8. This proves Conjecture 2.1 of Berkovich and Dhar in full. We further determine the limiting transition constants and a four-term asymptotic expansion for the transition centres. For n≥301, the signs in residue class $2$ are determined outside an interval of length 2εp/n2 centred at αp n2+βp n+γp+δp/n, where ε4=1500 and εp=150 for 5≤ p≤8, and all constants admit explicit analytic definitions. The proof combines a corrected saddle relation, higher-order expansions resolving cancellation between the two dominant saddle contributions, and a rescaled positive-integral argument that uniformly controls the small-degree range.
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BibTeX
@misc{jiang2026proof,
title = {{A proof of the Berkovich-Dhar conjecture modulo three}},
author = {Shutao Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08496},
eprint = {2610.08496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08496},
}
RIS
TY - UNPB
TI - A proof of the Berkovich-Dhar conjecture modulo three
AU - Shutao Jiang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.08496
UR - https://arxiv.org/abs/2610.08496
AB - Let P_n(q)=∏_j=1^n(1−q^3j−2)(1−q^3j−1). For each p∈{4,5,6,7,8}, we prove that the coefficients of P_n(q)^p in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coefficients are in fact strictly positive for 5≤ p≤8. This proves Conjecture 2.1 of Berkovich and Dhar in full. We further determine the limiting transition constants and a four-term asymptotic expansion for the transition centres. For n≥301, the signs in residue class $2$ are determined outside an interval of length 2ε_p/n^2 centred at α_p n^2+β_p n+γ_p+δ_p/n, where ε_4=1500 and ε_p=150 for 5≤ p≤8, and all constants admit explicit analytic definitions. The proof combines a corrected saddle relation, higher-order expansions resolving cancellation between the two dominant saddle contributions, and a rescaled positive-integral argument that uniformly controls the small-degree range.
ER -