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VII
Simple Analytic Proof of the

Prime Number Theorem

The magnificent Prime Number Theorem has received much atten-
tion and many proofs throughout the past century. If we ignore the
(beautiful) elementary proofs of Erd˝ os and Selberg and focus on the
analytic ones, we find that they all have some drawbacks. The origi-
nal proofs of Hadamard and de la Vall´ ee Poussin were based, to be
sure, on the nonvanishing of ζ(z) in z ≥ 1, but they also required
annoying estimates of ζ(z) at ∞, because the formulas for the coef-
ficients of the Dirichlet series involve integrals over infinite contours
(unlike the situation for power series) and so effective evaluation
requires estimates at ∞.
The more modern proofs, due to Wiener and Ikehara (and also
Heins) get around the necessity of estimating at ∞ and are indeed
based only on the appropriate nonvanishing of ζ(z), but they are
tied to certain results of Fourier transforms. We propose to return
to contour integral methods to avoid Fourier analysis and also to
use finite contours to avoid estimates at ∞. Of course certain errors
are introduced thereby, but the point is that these can be effectively
minimized by elementary arguments.
So let us begin with the well-known fact about the ζ-function (see
Chapter 6, page 60–61)

(z − 1)ζ(z) is analytic and zero-free throughout z ≥ 1. à 1)

This will be assumed throughout and will allow us to give our proof
of the Prime Number Theorem.

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