Distribution of the primes involving the ceiling function.

The distribution of the primes of the forms ⌊ n α ⌋ and ⌊ n α + β ⌋ are studied extensively, where ⌊ x ⌋ denotes the largest integer not exceeding x. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number 𝜃 with 0 < 𝜃 ≤ 1 , let π 𝜃 ′ (n) be the number of integers k with 1 ≤ k ≤ n 𝜃 such that ⌈ n / k ⌉ is prime and let π 𝜃 ′ ′ (n) be the number of primes p for which there exists an integer k with 1 ≤ k ≤ n 𝜃 such that p = ⌈ n / k ⌉ , where ⌈ x ⌉ denotes the least integer not less than x. These are closely related to the number of the prime factors of the denominator of the Bernoulli polynomial B n (x) − B n . In this paper, we study asymptotic properties of π 𝜃 ′ (n) and π 𝜃 ′ ′ (n). The methods in this paper are also effective for corresponding distribution functions of the primes involving the floor function. [ABSTRACT FROM AUTHOR]