Sieve methods and the twin prime conjecture

Abstract

For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $[ \; ]$ denote the floor or greatest integer function. For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime conjecture states that there are infinitely many prime numbers $p$ such that $p+2$ is also prime. In this paper we state a conjecture to the effect that given any integer $a>0$ there exists an integer $N_2(a)$ such that $$ \left[\frac{ap^2_{n+1}}{2(n+1)} \right] \leq \pi_2\left(p^2_{n+1} \right) $$ for all $n \geq N_2(a)$ and prove the conjecture in the case $a=1.$ This, in turn, establishes the twin prime conjecture.