A Positive Lower Bound for the Sum of Log-Reciprocal Twin Prime Products via Weighted Sieve

TL;DR

This paper refines the weighted sieve method to establish a positive lower bound for a sum over twin prime pairs, providing evidence supporting the twin prime conjecture by implying infinitely many such pairs.

Contribution

It introduces a novel weighted sieve approach to estimate sums over twin primes, advancing the analytical tools towards proving their infinitude.

Findings

Established a strict positive lower bound for the sum over twin prime pairs

Provided analytical evidence supporting the infinitude of twin primes

Enhanced the weighted sieve technique for prime gap analysis

Abstract

The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the weighted sieve method to estimate a sum over twin prime pairs, where each term is of the form \((1/p)(log(x^{{\alpha}}/p))^k\). We establish a strict positive lower bound for this sum, which implies the infinitude of twin prime pairs.