Distributional Wiener-Ikehara theorem and twin primes |
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Authors: | Jacob Korevaar |
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Institution: | KdV Institute ofMathematics, University of Amsterdam, Plantage Muidergracht 24,1018 TVAmsterdam, Netherlands |
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Abstract: | The Wiener-Ikehara theorem was devised to obtain a simple proof of the prime number theorem. It usesno other information about the zeta function ~(z) than that it is zero-free and analytic for Rez ? 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. |
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Keywords: | primary 40E05 secondary I IM45 11N05 42A38 44A10 461720 |
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