Convergence of the average of f(P1(n)) forces convergence of the average of f(P2(n)) to the same limit for every bounded f.
Tenenbaum, A rate estimate in Billingsley’s theorem for the size distribution of large prime factors, Quart
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Rigidity of Averages over the Two Largest Prime Factors
Convergence of the average of f(P1(n)) forces convergence of the average of f(P2(n)) to the same limit for every bounded f.