互反分拆的渐近公式
Asymptotic Formulae For Reciprocal Partitions
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中文总结 AI 辅助
该研究针对互反分拆计数函数s(n,m)，在m→∞时给出log s(n,m)的完整渐近展开，确定稀疏情形下乘法修正项层级，得到双参数一致渐近结果。
中文摘要 AI 辅助
对于整数1≤m≤n，令s(n,m)表示满足n=∑(j=1到m) k_j/j的非负整数m元组(k₁,…,kₘ)的个数。当m→∞时，对所有n≥m，我们得到log s(n,m)的完整渐近展开式，其系数由该问题留数结构导出的极限素块函数显式给出。我们还确定了稀疏情形下乘法修正项的完整层级，明确了各固定阶的显式常数，尤其是一阶修正常数1/4和(1−log2)/4。所得结果给出了当目标数n与允许的互反部分个数m同时增长时的双参数一致渐近式。
英文摘要
For integers $1\le m\le n$, let $s(n,m)$ denote the number of $m$-tuples $(k_1,\ldots,k_m)$ of nonnegative integers satisfying \[ n=\sum_{j=1}^{m}\frac{k_j}{j}. \] We obtain a complete asymptotic expansion for $\log s(n,m)$, uniformly for all $n\ge m$ as $m\to\infty$. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants $1/4$ and $(1-\log 2)/4$. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.
