Number Theory: An Introduction to Mathematics

3 Proof of the Prime Number Theorem 373

Proof Puts=σ+it,whereσandtare real, and let

g(σ,t)=−R{f′(s)/f(s)}.

Thus

g(σ,t)=

∫∞

0

e−σxcos(tx)dφ(x) forσ> 1.

Hence, by Schwarz’s inequality (Chapter I,§10),

g(σ,t)^2 ≤

∫∞

0

e−σxdφ(x)

∫∞

0

e−σxcos^2 (tx)dφ(x)

=g(σ, 0 )

∫∞

0

e−σx{ 1 +cos( 2 tx)}dφ(x)/ 2

=g(σ, 0 ){g(σ, 0 )+g(σ, 2 t)}/ 2.

Sincef(s)has a simple pole ats=1, by comparing the Laurent series off(s)and
f′(s)ats=1 (see Chapter I,§5) we see that

(σ− 1 )g(σ, 0 )→1asσ→ 1 +.

Similarly iff(s)has a zero of multiplicitym(t)≥0at1+it,wheret=0, then by
comparing the Taylor series off(s)andf′(s)ats= 1 +itwe see that

(σ− 1 )g(σ,t)→−m(t) asσ→ 1 +.

Thus if we multiply the inequality forg(σ,t)^2 by(σ− 1 )^2 and letσ→ 1 +, we obtain

m(t)^2 ≤{ 1 −m( 2 t)}/ 2 ≤ 1 / 2.

Therefore, sincem(t)is an integer,m(t)=0.

For f(s)=ζ(s), Proposition 7 gives the result of Hadamard and de la Vall ́ee
Poussin:

Corollary 8ζ( 1 +it)= 0 for every real t= 0.

The use of Schwarz’s inequality to prove Corollary 8 seems more natural than the
usual proof by means of the inequality 3+4cosθ+cos 2θ ≥0. It follows from
Corollary 8 that−ζ′(s)/ζ(s)− 1 /(s− 1 )is holomorphic in the closed half-plane
σ≥1. Hence, by (6), the hypotheses of the following theorem, due to Ikehara (1931),
are satisfied with

F(s)=−ζ′(s)/ζ(s), φ(x)=ψ(ex), h=A= 1.

Theorem 9Letφ(x)be a nondecreasing function for x≥ 0 such that the Laplace
transform

F(s)=

∫∞

0

e−sxdφ(x)