DISTRIBUTION OF THE PRIMESREPRESENTED BY (sic)x/n(sic) IN SHORT INTERVALS

Let (sic)x(sic) be the largest integer not exceeding x. For 0 < theta <= 1, let pi(theta) (x) denote the number of integers n with 1 <= n <= x(theta) such that (sic)x/n(sic) is prime. Recently, Ma, Chen and Wu obtained the interesting asymptotic formula pi(theta) (x) = x(0)/(1-theta)log x) + O(x(0) (log x)(-2)), provided that 23/47 < theta < 1. They further conjectured that this asymptotic formula can be extended to all 0 < theta < 1. In this paper, we give an improvement of their result by showing that 9/19 < theta < 1 is admissible.