On the Largest Prime Divisors of Numbers

The object of this note is to give an answer to the

QUERY: Let g(m) denote the largest prime divior of m. In what range does g(m) lie for all almost all values of m ≤ x?

More precisely, suppose that h(x) and H(x) aie two functions of x; let N(x) = N(h, H x) denote the number of numbers m ≤ x for which h(x) ≤ g(m) ≤ H(x). For what choices of h(x) and H(x) can we say that N(x)/x→I, as x→∞?.