Number Theory: An Introduction to Mathematics

3 Proof of the Prime Number Theorem for Arithmetic Progressions 403

3 Proof of the Prime Number Theorem for Arithmetic Progressions

The finite abelian group in which we are interested is the multiplicative groupZ×(m)of
integers relatively prime tom,wherem>1 will be fixed from now on. The group
Gm =Z×(m)has orderφ(m),whereφ(m)denotes as usual the number of positive
integers less thanmand relatively prime tom.
ADirichlet charactermodmis defined to be a functionχ :Z→Cwith the
properties

(i)χ(ab)=χ(a)χ(b)for alla,b∈Z,
(ii)χ(a)=χ(b)ifa≡bmodm,
(iii)χ(a)=0 if and only if(a,m)=1.

Any characterχofGmcan be extended to a Dirichlet character modmby putting
χ(a)=0ifa ∈Zand(a,m)=1. Conversely, on account of (ii), any Dirichlet
character modmuniquely determines a character ofGm.
To illustrate the definition, here are some examples of Dirichlet characters. In each
case we setχ(a)=0if(a,m)=1.

(I)m=pis an odd prime andχ(a)=(a/p)ifp
a,where(a/p)is the Legendre
symbol;
(II)m=4andχ(a)=1or−1 according asa≡1or−1 mod 4;
(III)m=8andχ(a)=1or−1 according asa≡±1or±3 mod 8.

We now return to the general case. By the results of the previous section we have

∑m

n= 1

χ(n)≡

{

φ(m) ifχ=χ 1 ,

0ifχ=χ 1 ,

and

∑

χ

χ(a)=

{

φ(m) ifa≡1modm,

0otherwise,

whereχruns through all Dirichlet characters modm.Furthermore

∑m

n= 1

χ(n)ψ(n)=

{

φ(m) ifχ=ψ,

0ifχ=ψ,

and

∑

χ

χ(a)χ( ̄b)=

{

φ(m) if(a,m)=1anda≡bmodm,

0otherwise.

Lemma 3If χ = χ 1 is a Dirichlet charactermodm then, for any positive
integer N ,

∣

∣

∣

∣

∑N

n= 1

χ(n)

∣

∣

∣

∣≤φ(m)/^2.