Power-Kernel Fluctuations, the Lambert W Subpolynomial Source, and a Local CRT Conditional Law
For a positive integer $n$ we define the power kernel $d_n = n/\operatorname{rad}(n)$ and the fluctuation $X_n = \log d_n$, where $\operatorname{rad}(n) = \prod_{p \mid n} p$. This paper develops three mutually supporting lines around this ``relative fluctuation''. First, the microscopic theory. From the kernel decomposition $n = v_d s$ ($s$ squarefree, $(s,d)=1$, $v_d = d \operatorname{rad}(d)$) we obtain the exact discrete law $\mathbb{P}(X_n = \log d) = \frac{6}{\pi^2} \frac{w(d)}{d \operatorname{rad}(d)}$, $w(d) := \prod_{p \mid d} \frac{p}{p+1}$, together with a three-branch…
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