Prime pairs and Zeta's zeros
Abstract
There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of pi2r(x), the number of prime pairs (p,p+2r) with p not exceeding x. However, it is still not known whether there are infinitely many prime pairs with given even difference! Using a strong hypothesis on (weighted) equidistribution of primes in arithmetic progressions, Goldston, Pintz and Yildirim have shown (2007) that there are infinitely many pairs of primes differing by at most sixteen. The present author uses a Tauberian approach to derive that the PPC is equivalent to specific boundary behavior of certain functions involving zeta's complex zeros. Under Riemann's Hypothesis and on the real axis, these functions resemble pair-correlation expressions. A speculative extension of Montgomery's classical work (1973) would imply that there must be an abundance of prime pairs.