AI 中文总结
本研究引入双变量生成函数与系数数组，建立Young格对角分拆与e^γ的Abel正则化多重zeta展开的双分次对应，给出钩型与非钩型贡献的规范分解及渐近结果。
AI 中文摘要
黎曼假设的一种等价表述最近催生了一种以Young格的对角为天然索引的分拆展开式。Segovia分离出这些对角上的钩型族$(r,1^m)$，并计算了它们的极限贡献$ρ_r$，同时发现非钩型族提供了缺失的贡献。\n我们引入了一个双变量有限生成函数，可同时打包每个固定超量对角上的所有Young形状。对每个固定的$r\geq1$，我们得到了对角生成多项式$D_r(n;z)$，并证明了\\[A_r(n)\sim C_r\\,n\log\log n\\]，其中$C_r$是一个显式收敛无穷乘积的第$(r-1)$个系数。此外，\\[C_r=\sum_{ν\vdash r-1} C_ν\\]给出了按超量$r-1$的分拆的规范分解。单分拆项贡献即为Segovia的钩常数$ρ_r$，而其余项同时给出了所有非钩型修正。\n随后我们通过引入系数$C_{r,d}$细化了这些常数，该系数同时记录Young格超量$r-1$和非单位行数$d$。行和可恢复固定超量常数$C_r$，而列和可恢复$e^γ$的Abel正则化多重zeta展开中的深度分解。更准确地说，每个分拆$ν\vdash r-1$都对应一个深度为$\ell(ν)$的Abel正则化多重zeta块。因此同一数组$(C_{r,d})$同时按Young格超量和多重zeta深度组织该分解。我们的结果关注该分解的组合与渐近结构，而非黎曼假设本身。
英文摘要
An equivalent formulation of the Riemann hypothesis recently led to a partition expansion naturally indexed by diagonals of the Young lattice. Segovia isolated the hook families $(r,1^m)$ on these diagonals and computed their limiting contributions $ρ_r$, while observing that non-hook families provide a missing contribution. We introduce a bivariate finite generating function that packages all Young shapes on every fixed-excess diagonal at once. For each fixed $r\geq1$, we obtain a diagonal generating polynomial $D_r(n;z)$ and prove \[ A_r(n)\sim C_r\,n\log\log n, \] where $C_r$ is the $(r-1)$st coefficient of an explicit convergent infinite product. Moreover, \[ C_r=\sum_{ν\vdash r-1} C_ν, \] giving a canonical decomposition over the partitions of the excess $r-1$. The one-part contribution is Segovia's hook constant $ρ_r$, while the remaining terms give all non-hook corrections simultaneously. We then refine these constants by introducing coefficients $C_{r,d}$ that record simultaneously the Young-lattice excess $r-1$ and the number $d$ of non-unit rows. Row sums recover the fixed-excess constants $C_r$, while column sums recover the depth decomposition in an Abel-regularized multiple-zeta expansion of $e^γ$. More precisely, each partition $ν\vdash r-1$ is identified with an Abel-regularized multiple-zeta block of depth $\ell(ν)$. Thus the same array $(C_{r,d})$ organizes the decomposition simultaneously by Young-lattice excess and multiple-zeta depth. Our results concern the combinatorial and asymptotic structure of this decomposition, rather than the Riemann hypothesis itself.
Comments33 pages, 1 figure
