The Wiener-Ikehara theorem was devised to obtain a simple proof of the prime number theorem. It uses no other information about the zeta function ζ(z) than that it is zero-free and analytic for Re z 1, apart from a simple pole at z = 1 with residue 1. In the Wiener-Ikehara theorem, the boundary behavior of a Laplace transform in the complex plane plays a crucial role. Subtracting the principal singularity, a first order pole, the classical theorem requires uniform convergence to a boundary function on every finite interval. Here it is shown that local pseudofunction boundary behavior, which allows mild singularities, is necessary and sufficient for the desired asymptotic relation. It follows that the twin-prime conjecture is equivalent to pseudofunction boundary behavior of a certain analytic function. Theorem 1.1. Let S(t) vanish for t < 0, be nondecreasing, continuous from the right and such that the Laplace transform MSC: primary 40E05; secondary 11M45, 11N05, 42A38, 44A10, 46F20
Abstract. 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 methods which permit simple derivations of the prime number theorem, associated with the labels LandauWiener-Ikehara and Newman. Other important areas in complex Tauberian theory are associated with the names Fatou-Riesz and Ingham. Recent refinements have been motivated by operator theory and include local H 1 and pseudofunction boundary behavior of transforms. Complex information has also led to better remainder estimates in connection with classical Tauberian theorems. Applications include the distribution of zeros and eigenvalues.
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