The prime-pair conjectures of Hardy and Littlewood

By (extended) Wiener-Ikehara theory, the prime-pair conjectures are equivalent to simple pole-type boundary behavior of corresponding Dirichlet series. Under a weak Riemann-type hypothesis, the boundary behavior of weighted sums of the Dirichlet series can be expressed in terms of the behavior of certain double sums View the MathML source. The latter involve the complex zeros of ζ(s) and depend in an essential way on their differences. Extended prime-pair conjectures are true if and only if the sums View the MathML source have good boundary behavior. Equivalently, a more general sum View the MathML source (with real ω>0) should have a boundary function (or distribution) that is well-behaved, apart from a pole R(ω)/(s−1/2) with residue R(ω) of period 2. [R(ω) could be determined for ω≤2.]