  There is extensive numerical support for the prime-pair conjecture (PPC) of
Hardy and Littlewood (1923) on the asymptotic behavior of pi_{2r}(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.
