On limit points of the sequence of normalized prime gaps

Abstract

Let $p_n$ denote the $n$th smallest prime number, and let $\mathbf{L}$ denote the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log p_n{\}}_{n=1}^{\infty }$ of normalized differences between consecutive primes. We show that for $k=9$ and for any sequence of $k$ nonnegative real numbers ${\beta }_1\le {\beta }_2\le ...\le {\beta }_k$, at least one of the numbers ${\beta }_j-{\beta }_i$ ($1\le i<j\le k$) belongs to $\mathbf{L}$. It follows at least $12.5$ of all nonnegative real numbers belong to $\mathbf{L}$.

Revised and improved

Cited by in corpus (5)

• Limit points of normalized prime gaps
• Positive density for consecutive runs of sums of two squares
• Prime Tuples and Siegel Zeros
• Distance between natural numbers based on their prime signature
• Small gaps between almost-twin primes