Year: 2026
Type: preprint
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 $n\leq x$ such that $\#([n, n+f(n)] \cap \mathbb{P})=m$ for sufficiently large $x$, where $\mathbb{P}$ 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.
Source: arXiv (Cornell University)
Author: Sono, Keiju
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: L関数と篩法による素数分布の研究
OpenAlex ID: W7213517366
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