Distribution of the primes involving the ceiling function

作者：Ma, Wu-Xia^*; Chen, Yong-Gao; Wu, Bing-Ling

来源：International Journal of Number Theory, 2019, 15(3): 597-611.

DOI：10.1142/S1793042119500313

• Distribution of the primes
• the ceiling function
• asymptotic formula
• Bernoulli polynomials

The distribution of the primes of the forms left perpendicular n(alpha) right perpendicular and left perpendicular n alpha + beta right perpendicular are studied extensively, where left perpendicular x right perpendicular 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 theta with 0 < theta <= 1, let pi'(theta)(n) be the number of integers k with 1 <= k <= n(theta) such that inverted right perpendicular n/k inverted left perpendicular is prime and let pi ''(theta)(n) be the number of primes p for which there exists an integer k with 1 <= k <= n(theta) such that p = inverted right perpendicular n/k inverted left perpendicular, where inverted right perpendicular x inverted left perpendicular 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 pi'(theta)(n) and pi ''(theta)(n). The methods in this paper are also effective for corresponding distribution functions of the primes involving the floor function.

• 出版日期2019-4
• 单位南京师范大学