Goldbach and Twin Prime Pairs: A Sieve Method to Connect the Two

TL;DR

This paper introduces a sieve method based on interval divisibility properties to establish lower bounds for twin primes and Goldbach pairs, linking the Goldbach conjecture to the Riemann hypothesis.

Contribution

It presents a novel elementary sieve approach that connects twin primes and Goldbach pairs, and demonstrates the conditional proof of Goldbach's conjecture assuming the Riemann hypothesis.

Findings

Established lower bounds for twin primes and Goldbach pairs.

Demonstrated the method's efficacy in relating Goldbach's conjecture to the Riemann hypothesis.

Used prime distribution formulas to support bounds.

Abstract

This paper proposes, and demonstrates the efficacy of, an elementary method for establishing a lower bound for cardinalities of selected sets of twin primes, and shows that the proof employed may be modified for selected sets of Goldbach pairs. Our sieve method is centred on the restrictive properties of intervals, specifically regarding divisibility distributions. We implicitly use the Chinese Remainder Theorem by way of the use of the midpoint in our intervals, and consider the sieve of Eratosthenes in such a way as to find a set of primes whose distribution is mirror-symmetrical about that midpoint. Bounds are established through the use of the formulae closely associated with the Prime Number theorem and the Mertens theorem. We show that the Goldbach conjecture is true if the Riemann hypothesis is true.