PreprintarXiv licencearXiv
Bounded intervals containing a given number of primes
Abstract
Let m be a non-negative integer and f(x) a positive, non-decreasing function satisfying certain conditions. We give an explicit lower bound for the number of integers n≤ x such that #([n, n+f(n)] ∩ ℙ)=m for sufficiently large x, where ℙ denotes the set of prime numbers. This work extends the results of Mastrostefano and of Freiberg, and also makes the lower bound of Masrtrostefano explicit. In addition, we show that if the interval length is sufficiently large, then there exist infinitely many bounded intervals of the same length that contain exactly a prescribed number of primes.
Related
- Triangles minimize the Monge-Ampère eigenvalue among planar bounded convex domains of a given area
- Biases in the distribution of primes in short intervals
- Quasimodular Forms, Partitions, and Primes in Intervals
- The Poisson Tail Conjecture for primes in short intervals
- Primes in short intervals assuming the Riemann hypothesis
BibTeX
@misc{sono2026bounded,
title = {{Bounded intervals containing a given number of primes}},
author = {Keiju Sono},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18414},
eprint = {2609.18414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18414},
}
RIS
TY - UNPB
TI - Bounded intervals containing a given number of primes
AU - Keiju Sono
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18414
UR - https://arxiv.org/abs/2609.18414
AB - Let m be a non-negative integer and f(x) a positive, non-decreasing function satisfying certain conditions. We give an explicit lower bound for the number of integers n≤ x such that #([n, n+f(n)] ∩ ℙ)=m for sufficiently large x, where ℙ denotes the set of prime numbers. This work extends the results of Mastrostefano and of Freiberg, and also makes the lower bound of Masrtrostefano explicit. In addition, we show that if the interval length is sufficiently large, then there exist infinitely many bounded intervals of the same length that contain exactly a prescribed number of primes.
ER -