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 Σ2k∗(s). The latter involve the complex zeros of ζ(s)ζ(s) and depend in an essential way on their differences. Extended prime-pair conjectures are true if and only if the sums Σ2k∗(s) have good boundary behavior. Equivalently, a more general sum Σω∗(s) (with real ω>0ω>0) should have a boundary function (or distribution) that is well-behaved, apart from a pole R(ω)/(s−1/2)R(ω)/(s−1/2) with residue R(ω)R(ω) of period 22. [R(ω)R(ω) could be determined for ω≤2ω≤2.]