# On the Distribution of Twin Primes

## TL;DR

This paper introduces a new sieve method for counting twin primes up to a certain range, providing asymptotic bounds and insights into their distribution, which could lead to progress on prime number conjectures.

## Contribution

It develops a novel sieve approach dependent on a parameter, offering a new perspective on the distribution and asymptotic bounds of twin primes.

## Key findings

- Established an asymptotic bound for twin primes less than x
- Introduced a fundamental structure for twin prime distribution
- Proposed a variant of the prime number theorem

## Abstract

We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter ${\lambda}_x$ and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of ${\lambda}_x$, establishes venues, conducive to a discovery of a partial result that can be considered as a suitable variant of the prime number theorem. Furthermore, we obtain an asymptotic bound of the number of twin primes less than $x$.