关于前两大素因子平均值的刚性
Rigidity of Averages over the Two Largest Prime Factors
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中文总结 AI 辅助
该研究解决了Alladi与Johnson关于数论中前两大素因子平均值的问题，证明若最大素因子对应加权平均收敛，则第二大素因子对应加权平均必收敛到同一极限，借助Dickman核与Wiener陶伯定理完成推导。
中文摘要 AI 辅助
设\\(P_1(n)\\)和\\(P_2(n)\\)分别为\n\n的最大和第二大不同素因子。Alladi与Johnson提出问题：是否存在定义在素数上的有界函数\f\f，使得极限\f\frac{1}{x}\sum_{2\le n\le x} f(P_1(n)) \longrightarrow κ_1\f\f与\f\frac{1}{x}\sum_{2\le n\le x} f(P_2(n)) \longrightarrow κ_2\f\f均存在且\f\fκ_1≠κ_2\f\f，其中当\f\n\f为素数幂时令\f\ff(P_2(n))=0\f\f。我们证明这是不可能的：第一个平均值的收敛会迫使第二个平均值收敛到同一极限。\n在\f\f\log\log\f\f尺度下，两个平均值可表示为与显式Dickman核（迪克曼核）的卷积。与\f\fP_1\f\f-平均值相关的Dickman核的傅里叶变换无实零点。随后Wiener的Tauberian定理（维纳陶伯定理）给出与加权素数和相关的平移测度的淡收敛，进而得到第二个平均值的收敛性。
英文摘要
Let \(P_1(n)\) and \(P_2(n)\) be the largest and second-largest distinct prime factors of \(n\), respectively. Alladi and Johnson asked whether there exists a bounded function \(f\) on the primes for which both limits \(\frac{1}{x}\sum_{2\le n\le x} f(P_1(n)) \longrightarrow κ_1\) and \(\frac{1}{x}\sum_{2\le n\le x} f(P_2(n)) \longrightarrow κ_2\) exist with \(κ_1\neqκ_2\), where we set \(f(P_2(n))=0\) when \(n\) is a prime power. We prove that this is impossible: convergence of the first average forces convergence of the second to the same limit. On the \(\log\log\)-scale, the two averages are expressed as convolutions with explicit Dickman kernels. The Fourier transform of the Dickman kernel associated with the \(P_1\)-average has no real zeros. Wiener's Tauberian theorem then yields vague convergence of the translated measures associated with the weighted prime sums. The convergence of the second average follows.
