模5下Berkovich-Dhar猜想的一个证明
A proof of the Berkovich-Dhar conjecture modulo five
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中文总结 AI 辅助
该研究证明了模5下Berkovich-Dhar猜想的符号模式与极限转移断言，确定了相关转移常数，修正了数值估计，通过多种技术完成了对所有正整数的证明。
中文摘要 AI 辅助
我们证明了模5下Berkovich-Dhar猜想的符号模式与极限转移断言，涵盖有限Borwein乘积的平方和立方两种幂次。在每种幂次中，剩余类0的系数严格为正，而剩余类3和4的序列在剔除零项后恰好有一次正到负的符号变化。我们确定了全部四个转移常数，修正了猜想中的初步数值估计，并给出带有明确有界误差的线性修正项。一个共同的四弧展开式适用于两种幂次。在每个消失振幅附近，加权相邻差的严格负性决定了转移。立方情形的无穷乘积中还有两个恒等消失的分量；移位鞍点展开式可解决由此产生的有限乘积边界项。正级数恒等式、一致余项估计和精确整数递推关系完成了对所有正整数的证明。
英文摘要
We prove the sign-pattern and limiting-transition assertions of the Berkovich--Dhar conjecture modulo five, covering both the square and the cube of the finite Borwein product. In each power, the coefficients in residue class zero are strictly positive, while the sequences in residue classes three and four have exactly one positive-to-negative sign change after zero terms are omitted. We determine all four transition constants, correct the preliminary numerical estimates in the conjecture, and give their linear corrections with explicit bounded errors. A common four-arc expansion governs both powers. Near each vanishing amplitude, strict negativity of a weighted adjacent difference determines the transition. The cubic case also has two identically vanishing components in the infinite product; a shifted saddle expansion resolves the resulting finite-product boundary terms. Positive series identities, uniform remainder estimates, and an exact integer recurrence complete the proof for every positive integer.
发表机构
- East China Normal University（华东师范大学）
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