Distribution of the values of Euler's \(\varphi\)-function and of the sum-of-divisors function on integers free of large prime factors (Q2714194)

For a multiplicative function \(f(n)\) set \(F(x, y)\) for the number of integers \(n\) for which \(f(n)\leq x\) and \(P(n)\leq y\), where \(P(n)\) is the largest prime factor of \(n\). The author develops an asymptotic formula for \(F(x, y)\) under some restrictions on the growth of \(f(n)\). The result generalizes known formulas for specific \(f(n)\), and generalizes asymptotic formulas in which \(P(n)\) is not restricted.