Number Theory: An Introduction to Mathematics

408 X A Character Study

Proposition 8

∑

n≤xχ^1 (n)Λ(n)∼x,

∑

n≤xχ(n)Λ(n)=o(x)ifχ=χ^1.

Proof For any Dirichlet characterχ, put

g(s)=−ζ′(s)/ζ(s)−L′(s,χ)/ 2 L(s,χ)−L′(s,χ)/ ̄ 2 L(s,χ), ̄

h(s)=−ζ′(s)/ζ(s)−L′(s,χ)/ 2 iL(s,χ)+L′(s,χ)/ ̄ 2 iL(s,χ). ̄

Forσ=Rs>1wehave

g(s)=

∑∞

n= 1

{ 1 +Rχ(n)}Λ(n)n−s,

h(s)=

∑∞

n= 1

{ 1 +Iχ(n)}Λ(n)n−s.

Ifχ=χ 1 then, by Proposition 7,g(s)− 1 /(s− 1 )andh(s)− 1 /(s− 1 )are holomor-
phic forRs≥1. Since the coefficients of the Dirichlet series forg(s)andh(s)are
nonnegative, it follows from Ikehara’s theorem (Theorem IX.9) that

∑

n≤x

{ 1 +Rχ(n)}Λ(n)∼x,

∑

n≤x

{ 1 +Iχ(n)}Λ(n)∼x.

On the other hand, ifχ=χ 1 theng(s)− 2 /(s− 1 )andh(s)− 1 /(s− 1 )are holomorphic
forRs≥1, from which we obtain in the same way

∑

n≤x

{ 1 +χ 1 (n)}Λ(n)∼ 2 x,

∑

n≤x

Λ(n)∼x.

The result follows.

The prime number theorem for arithmetic progressions can now be deduced
immediately. For, bythe orthogonality relations and Proposition 8, if 1≤a <m
and(a,m)=1, then

ψ(x;m,a)=

∑

n≤x,n≡amodm

Λ(n)

=

∑

χ

χ( ̄a)

∑

n≤x

χ(n)Λ(n)/φ(m)

∼x/φ(m).

It is also possible to obtain error bounds in the prime number theorem for arith-
metic progressions of the same type as those in the ordinary prime number theorem.
For example, it may be shown that for eachα>0,