Sieves for Twin Primes in Class I

TL;DR

This paper develops specialized sieves for identifying twin primes of a specific form (class I) and analyzes their distribution, revealing regularities and asymptotic behaviors related to their ranks and non-ranks.

Contribution

It introduces new sieve constructions for twin primes in class I, characterizes their ranks, and analyzes their asymptotic distribution considering the parameter D.

Findings

Sieves for twin primes in class I are constructed.

Regularities in non-ranks help determine the number of twin-D-I ranks.

Asymptotic form of the main term depends on D.

Abstract

Sieves are constructed for twin primes in class I, which are of the form 2m+/-D, D>=3 odd. They are characterized by their twin-D-I rank m. They have no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin-D-I ranks and non-ranks make up the set of positive integers. Regularities of non-ranks allow obtaining the number of twin-D-I ranks. It involves considerable cancellations so that the asymptotic form of its main term collapses to the expected form, but its coefficient depends on D.