The Wiener–ikehara Theorem by Complex Analysis
نویسندگان
چکیده
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?
منابع مشابه
Distributional Wiener–Ikehara theorem and twin primes
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 s...
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