DISTRIBUTION OF THE PRIMES REPRESENTED BY ⌊ xn ⌋ IN SHORT INTERVALS

Let ⌊x⌋ be the largest integer not exceeding x. For 0<𝜃≤1, let π𝜃(x) denote the number of integers n with 1≤n≤x𝜃 such that ⌊x∕n⌋ is prime. Recently, Ma, Chen and Wu obtained the interesting asymptotic formula π𝜃(x)=x𝜃(1−𝜃)log⁡x+O(x𝜃(log⁡x)−2), provided that 2347<𝜃<1. They further conjectured that this asymptotic formula can be extended to all 0<𝜃<1. In this paper, we give an improvement of their result by showing that 919<𝜃<1 is admissible.