where ζ is the Riemann zeta-function and s > 1. It follows from (1.2) that for any positive integer k, dk(n) is the number of ordered k-tuples (n1, . . . , nk) of positive integers such that n1 . . . nk = n. In particular, d2(n) is the number of distinct positive divisors of n. For real z, x, w, define ∆z(x,w) = #{n ≤ x : dz(n) > w}, (1.3) ∆z(x,w) = #{n ≤ x : dz(n) ≥ w}, (1.4) where #B means the number of members of the finite set B (note that ∆z(x,w) ≤ ∆z(x,w)). Our main objective is to obtain good upper bounds for ∆z(x,w) and good lower bounds for ∆z(x,w) when z > 1, x is large, and logw is larger than the normal order of log dz(n) for n ≤ x. Before stating our results, we must specify some notation. Unless otherwise stated, r, t, u, v, w, x, y, z, α, β, δ, e denote real numbers, with e > 0. (For consistency with the notation of some earlier authors, we shall let y denote a positive integer in Section 3.) We use γ to denote Euler’s constant, while k, m, n represent positive integers and p is a (positive) prime number. If a is a nonnegative integer, p ‖n means that p |n and