On a Restricted Divisor Problem

Let 0 < α < 1/2 and let d α (n) be the number of positive divisors k of n such that n α ≤ k ≤ n 1-α , which we call a restricted divisor function. In the case α = 1/N (N ∈ N) we derive an asymptotic representation of Σn≤x d α (n) . Furthermore we study the mean square of P α (x) = Σ l≤x α φ (x/l), which seems to be a natural object in the case of a restricted divisor problem.