Jaap Korevaar

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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)
was surmised already by Legendre and Gauss. However, it took a hundred years before the first proofs appeared, one by Hadamard and one by de la Vall~e Poussin (1896). Their and all but one of the subsequent proofs make heavy use of the Riemann zeta function. (The one exception is the long so-called elementary proof by Selberg [11] and Erd6s [41.) For Re s >(More)
Taking r > 0, let π2r(x) denote the number of prime pairs (p, p+ 2r) with p ≤ x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r > 0. There seems to be no good conjecture for the remainders ω2r(x) = π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)− li(x). However,(More)
For k > 1, r 6= 0 and large x, let π 2r (x) denote the number of prime pairs (p, p+2r) with p ≤ x. By the Bateman–Horn conjecture the function π 2r(x) should be asymptotic to (2/k)C k 2rli2(x), with certain specific constants C 2r . Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy–Littlewood constants(More)
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