Limit Points of the Sequence of Normalized Differences Between Consecutive Prime Numbers

Daniel A. Goldston () and Andrew H. Ledoan ()
Additional contact information
Daniel A. Goldston: San José State University, Department of Mathematics
Andrew H. Ledoan: University of Tennessee at Chattanooga, Department of Mathematics

A chapter in Analytic Number Theory, 2015, pp 115-125 from Springer

Abstract: Abstract Let p n denote the nth prime number and let d n = p n + 1 − p n $$d_{n} = p_{n+1} - p_{n}$$ denote the nth difference in the sequence of prime numbers. Erdős and Ricci independently proved that the set of limit points of d n log p n $$\frac{d_{n}} {\log p_{n}}$$ , the normalized differences between consecutive prime numbers, forms a set of positive Lebesgue measure. Hildebrand and Maier answered a question of Erdős and proved that the Lebesgue measure of the set of limit points of d n log p n $$\frac{d_{n}} {\log p_{n}}$$ in the interval [0, T] is ≫ T as T → ∞ $$T \rightarrow \infty$$ . Currently, the only specific limit points known are 0 and ∞ $$\infty$$ . In this note, we use the method of Erdős to obtain specific intervals within which a positive Lebesgue measure of limit points exist. For example, the intervals 1 8 , 2 $$\left [\frac{1} {8},2\right ]$$ and 1 40 , 1 $$\left [ \frac{1} {40},1\right ]$$ both have a positive Lebesgue measure of limit points.

Keywords: Hardy–Littlewood prime k-tuple conjecture; Limit points of normalized differences between consecutive prime numbers; Singular series (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-22240-0_8

Ordering information: This item can be ordered from
http://www.springer.com/9783319222400

DOI: 10.1007/978-3-319-22240-0_8

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().