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

TL;DR

This paper investigates the asymptotic density of prime k-tuples, computes specific Skewes numbers for various tuples, and supports the Hardy-Littlewood conjecture with numerical data and algorithms.

Contribution

It extends the computation of Skewes numbers to eight additional prime k-tuples and offers algorithms to determine these numbers, supporting the Hardy-Littlewood conjecture.

Findings

Computed Skewes numbers for 8 prime k-tuples

Provided numerical evidence supporting Hardy-Littlewood conjecture

Developed algorithms for calculating Skewes numbers

Abstract

In 1922 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.