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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)
The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman's contour integration method can be adapted to establish the Wiener–Ikehara theorem. A simple special case suffices for the PNT. But what about the twin-prime problem?
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)
The prime number theorem provided the chief impulse for complex Tauberian theory, in which the boundary behavior of a transform in the complex plane plays a crucial role. We consider Laplace transforms of bounded functions. Our Tauberian theorem does not allow first-order poles on the imaginary axis, but any milder singularities, characterized by… (More)
There are parallels between de Bruijn's early work in analysis and that of the author. However, Dick's work soon became much broader and deeper. While the present paper reviews several topics of common interest, its main content is a short version of Dick's important article related to Riemann's Hypothesis, entitled 'The roots of trigonometric integrals'… (More)