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Complex-analytic and related boundary properties of transforms give information on the behavior of pre-images. The transforms may be power series, Dirichlet series or Laplace-type integrals; the pre-images are series (of numbers) or functions. The chief impulse for complex Tauberian theory came from number theory. The first part of the survey emphasizes(More)
CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. Abstract. Taking r > 0, let π 2r (x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The prime-pair conjecture of Hardy and Little-wood (1923) asserts that π 2r (x) ∼ 2C 2r li 2 (x) with an(More)
For k > 1, r = 0 and large x, let π k 2r (x) denote the number of prime pairs (p, p k + 2r) with p ≤ x. By the Bateman–Horn conjecture the function π k 2r (x) should be asymptotic to (2/k)C k 2r li 2 (x), with certain specific constants C k 2r. Heuristic arguments lead to the conjecture that these constants have mean value one, just like the(More)
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(More)
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