发表机构
Mutah University（穆塔赫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了分拆理论密度公式中，级数趋向对应密度的条件，修正了相关已发表论断，为素数Chebotarev集提供了Alladi定理的分拆类似物。
AI 中文摘要
小野（Ono）、施奈德（Schneider）和瓦格纳（Wagner）发现了Alladi对偶公式的分拆理论类似物：对于许多自然数集S，当q→1时，对最小部分属于S的不同部分分拆求和的级数-∑μ_P(λ)q^|λ|趋向于S的密度。我们确定这种情况发生的条件。径向情况下，求和核为Gumbel密度，Wiener定理表明，对于有界函数f，当且仅当f在每个区间(x, x+cx/log x]上的均值为L时，相关级数趋向于L；自然密度并不充分。将该级数乘以log(1/(1-q))，当且仅当S在每个此类区间中有(cδ+o(1))x/(log x)^2个元素时，该级数趋向于δ；这为素数的Chebotarev集提供了Alladi定理的分拆类似物。在实轴外，最大权重增长为e^{Ψ_β/t}，其中Ψ_β>0。沿q=e^{-t(1+iβ)}，当且仅当|β|<β_m时，模m≥3的剩余类公式成立，β_m为显式Clausen函数阈值且β_m→0，对于有界偏差的随机集，该公式几乎必然不成立。这些结果修正了若干已发表的论断。
英文摘要
Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\subseteq\N$, the series $-\sumμ_{\mathcal P}(λ)q^{|λ|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to1$. We determine when this happens. Radially, the summation kernel is a Gumbel density, and Wiener's theorem shows that for bounded $f$ the associated series tends to $L$ if and only if $f$ has mean $L$ on every interval $(x,x+cx/\log x]$; natural density does not suffice. Multiplied by $\log(1/(1-q))$, the series tends to $δ$ if and only if $S$ has $(cδ+o(1))x/(\log x)^2$ elements in each such interval; this gives a partition analogue of Alladi's theorem for Chebotarev sets of primes. Off the real axis the largest weights grow like $e^{Ψ_β/t}$ with $Ψ_β>0$. Along $q=e^{-t(1+iβ)}$, the formula for residue classes modulo $m\ge3$ holds exactly when $|β|<β_m$, an explicit Clausen-function threshold with $β_m\to0$, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.
