A Note on the Distribution of Primes in Intervals

Tristan Freiberg ()
Additional contact information
Tristan Freiberg: University of Waterloo, Department of Pure Mathematics

A chapter in Irregularities in the Distribution of Prime Numbers, 2018, pp 23-44 from Springer

Abstract: Abstract Assuming a certain form of the Hardy–Littlewood prime tuples conjecture, we show that, given any positive numbers λ 1, …, λ r and nonnegative integers m 1, …, m r, the proportion of positive integers n ≤ x $$n {\leqslant } x$$ for which, for each j ≤ r $$j {\leqslant } r$$ , the interval ( n , n + ( λ 1 + ⋯ + λ j ) log x ] $$(n, n + (\lambda _1 + \cdots + \lambda _j)\log x]$$ contains exactly m 1 + ⋯ + m j primes, is asymptotically equal to ∏ j = 1 r ( e − λ j λ m j ∕ m j ! ) $$\prod _{j = 1}^r( {\mathrm {e}}^{-\lambda _j}\lambda ^{m_j}/m_j!)$$ as x →∞. This extends a result of Gallagher, who considered the case r = 1. We use a direct inclusion–exclusion argument in place of Gallagher’s moment calculation, thereby avoiding recourse to the moment determinacy property of Poisson distributions.

Date: 2018
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:978-3-319-92777-0_2

Ordering information: This item can be ordered from
http://www.springer.com/9783319927770

DOI: 10.1007/978-3-319-92777-0_2

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 ().