Rigidity of Averages over the Two Largest Prime Factors

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 $\frac{1}{x}\sum_{2\le n\le x} f(P_1(n)) \longrightarrow \kappa_1$ and $\frac{1}{x}\sum_{2\le n\le x} f(P_2(n)) \longrightarrow \kappa_2$ exist with $\kappa_1\neq\kappa_2$, 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.