The Hardy-Littlewood conjectures on the twin primes and the binary Goldbach problem are true

2020/06/04 by Maurizio Laporta, Laporta, Maurizio
Mathematics · #11N05 #11N37 #11P32 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #General Mathematics (math.GM) #Limits and Structures in Graph Theory #math.GM #msc:11N05 #msc:11N37 #msc:11P32

paper · pdf · doi:10.48550/arxiv.2006.04547

Lemma 1 is wrong

openalex publication_date 2020/06/04 · openalex created_date 2020/06/12 · arxiv created 2020/08/28 · arxiv updated 2020/08/31 · openalex updated_date 2026/07/28

Abstract

A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting function expressed in terms of the von Mangoldt function and its incomplete form. In a completely analogous way, we solve the conjugate conjecture on the representations of any even integer as the sum of two prime numbers.

Related

• Some remarks on the first Hardy-Littlewood conjecture
• On Ramanujan expansions and primes in arithmetic progressions
• On Chen's theorem, Goldbach's conjecture and almost prime twins II
• Some Results on a Conjecture of Hardy and Littlewood
• A Hypothesis on Upper Bound of Goldbach Counting Function