Distributional Wiener-Ikehara theorem and twin primes

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 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.