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for some absolute constant c1, uniformly for all complex-valued additive functions f(n) and real x ≥ 2. (For the relevant literature on the Turán– Kubilius inequality, see Elliott [4]; in particular Chapter 4.) Let P (n) denote the largest prime factor of n if n > 1, and P (1) = 1. Let S(x, y) = {n | 1 ≤ n ≤ x, P (n) ≤ y}. In 1982, Alladi [1] obtained a Turán–Kubilius inequality for integers n which have no large prime factors. More specifically, he showed that the inequality is valid uniformly for all strongly additive functions f(n) and n ∈ S(x, y), where x, y satisfies x ≥ 20, and exp{(log x)} ≤ y ≤ x. In [8], we further extended this range to y ≥ (log x). In the present paper we derive a Turán–Kubilius inequality for (general) additive functions f(n), where n ∈ S(x, y). We require uniform asymptotic estimates for Ψ(x, y), where

The Turán-Kubilius inequality for integers free of large prime factors (II)