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 Σ 2k(s). 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 Σ 2k(s) have good boundary behavior. Equivalently, a more general sum Σ ω(s) (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.] c ⃝ 2011 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.