Analog of the Skewes number for twin primes

2007/07/06 by Marek Wolf, Wolf, Marek · 2 citations
Mathematics · Psychology · #Advanced Mathematical Identities #Analytic Number Theory Research #Arithmetic #FOS: Mathematics #Finite Group Theory Research #Mathematics #Number Theory (math.NT) #Psychology #math.NT

paper · pdf · doi:10.48550/arxiv.0707.0980

published in arXiv (Cornell University) (Cornell University) · Changes: New Figure 1 and a few sentences of justification in favor of the conjecture (5) are made

openalex publication_date 2007/07/06 · arxiv created 2008/01/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The results of the computer investigation of the sign changes of the difference between the number of twin primes π₂(x) and the Hardy--Littlewood conjecture c₂\Li₂(x) are reported. It turns out that π₂(x) - c₂\Li₂(x) changes the sign at unexpectedly low values of x and for x<2⁴² there are over 90000 sign changes of this difference. It is conjectured that the number of sign changes of π₂(x) - c₂\Li₂(x) for x∈ (1, T) is given by √ T/log(T).

Citations

• On the difference π(x) - li(x)
• A new bound for the smallest 𝑥 with 𝜋(𝑥)>fc(r)
• 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 )-li( x ) (I)
• On the sign of the difference 𝜋(𝑥)-𝑙𝑖(𝑥)

Related

• The Skewes Number for Twin Primes: Counting Sign Changes of π 2(x) – C2Li2(x)
• Some history about Twin Prime Conjecture
• The twin prime conjecture
• On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood
• On the Twin Prime Conjecture