- Distributional Wiener-Ikehara theorem and twin primes.
- Indagationes Mathematicae
- Volume | Issue number
- 16 | 1
- Number of pages
- Document type
- Faculty of Science (FNWI)
- Korteweg-de Vries Institute for Mathematics (KdVI)
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 zeta (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.
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