On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood

2019/01/01 by László Tóth · 3 citations
Mathematics · #Analytic Number Theory Research #Mathematics and Applications #Limits and Structures in Graph Theory #Tuple #Twin prime #Conjecture #Prime (order theory) #Mathematics #Combinatorics #Prime number #Natural density #Discrete mathematics

paper · pdf · doi:10.12921/cmst.2019.0000033

openalex publication_date 2019/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06

Abstract

Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011, Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find "Skewes numbers" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.

Citations

• Sur la distribution des nombres premiers
• Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes
• On the Difference π( x ) − lix (II)

Cited by

• Cousin prime [wikipedia]
• Prime quadruplet [wikipedia]
• Skewes's number [wikipedia]

Related

• Some remarks on the first Hardy-Littlewood conjecture
• On the differences between consecutive prime numbers, I
• Asymptotic of the greatest distance between adjacent primes and the Hardy-Littlewood conjecture
• Analog of the Skewes number for twin primes
• On the frequency of small gaps between the primes