Number Theory: An Introduction to Mathematics

3 Proof of the Prime Number Theorem for Arithmetic Progressions 405

Proposition 4L(s,χ)=Πp( 1 −χ(p)p−s)−^1 forσ> 1 , where the product is taken
over all primes p.

Proof The propertyχ(ab)=χ(a)χ(b)for alla,b∈Nenables the proof of Euler’s
product formula forζ(s)to be carried over to the present case. Forσ>0wehave

( 1 −χ(p)p−s)−^1 = 1 +χ(p)p−s+χ(p^2 )p−^2 s+χ(p^3 )p−^3 s+···

and hence forσ> 1
∏

p≤x

( 1 −χ(p)p−s)−^1 =

∑

n≤Nx

χ(n)n−s,

whereNxis the set of all positive integers whose prime factors are all≤x. Letting
x→∞, we obtain the result.

It follows at once that

L(s,χ 1 )=ζ(s)

∏

p|m

( 1 −p−s)

and that, for any Dirichlet characterχ,L(s,χ)=0forσ>1.

Proposition 5−L′(s,χ)/L(s,χ)=

∑∞

n= 1 χ(n)Λ(n)/n

sforσ> 1.

Proof The seriesω(s,χ)=

∑∞

n= 1 χ(n)Λ(n)n

−sconverges absolutely and uniformly

in any half-planeσ ≥ 1 +ε,whereε>0. Moreover, as in the proof of Proposi-
tion IX.6,

L(s,χ)ω(s,χ)=

∑∞

j= 1

χ(j)j−s

∑∞

k= 1

χ(k)Λ(k)k−s=

∑∞

n= 1

n−s

∑

jk=n

χ(j)χ(k)Λ(k)

=

∑∞

n= 1

n−sχ(n)

∑

d|n

Λ(d)=

∑∞

n= 1

n−sχ(n)logn=−L′(s,χ). 

As in the proof of Proposition IX.6, we can also prove directly thatL(s,χ)= 0
forσ>1, and thus make the proof of the prime number theorem for arithmetic
progressions independent of Proposition 4.
The following general result, due to Landau (1905), considerably simplifies the
subsequent argument (and has other applications).

Proposition 6Letφ(x)be a nondecreasing function for x≥ 0 such that the integral

f(s)=

∫∞

0

e−sxdφ(x) (†)

is convergent forRs>β. Thus f is holomorphic in this half-plane. If the definition
of f can be extended so that it is holomorphic on the real segment(α,β], then the
integral in(†)is convergent also forRs>α. Thus f is actually holomorphic, and(†)
holds, in this larger half-plane.