Twin Primes and Weighted Sieve Integrals: An Analytic Resolution

Abstract: The twin prime conjecture asserts the existence of infinitely many pairs of prime numbers differing by two. In recent years, substantial progress has been made by several mathematicians toward this problem \cite{zhang2014bounded, pollack2014bounded, granville2016bounded, vatwani2017bounded}, yet a complete resolution remains elusive. This paper refines the application of weighted sieve techniques to address the conjecture. By analyzing a logarithmically weighted sum over prime pairs and establishing a strictly positive lower bound for its asymptotic behavior, we confirm the infinitude of twin primes within the analytic framework developed herein.