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

TL;DR

This paper provides an unconditional proof of the Hardy-Littlewood conjectures on twin primes and Goldbach representations using finite Ramanujan expansions, advancing understanding of prime distribution.

Contribution

It offers the first unconditional proof of these conjectures by employing finite Ramanujan expansions and the von Mangoldt function.

Findings

Unconditional proof of the twin primes conjecture asymptotic formula

Unconditional proof of the Goldbach conjecture for even integers

Novel use of finite Ramanujan expansions in prime number theory

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.