Number Theory: An Introduction to Mathematics

3 Proof of the Prime Number Theorem for Arithmetic Progressions 409

ψ(x;m,a)=x/φ(m)+O(x/logαx),

π(x;m,a)=Li(x)/φ(m)+O(x/logαx),

where the constants implied by theO-symbols depend onα, but not onmora.
In the same manner as for the Riemann zeta-functionζ(s)it may be shown that the
DirichletL-functionL(s,χ)satisfies a functional equation, providedχis a primitive
character. (Here a Dirichlet characterχmodmisprimitiveif for each proper divisord
ofmthere exists an integera≡1moddwith(a,m)=1andχ(a)=1.) Explicitly,
ifχis a primitive character modmand if one puts

Λ(s,χ)=(m/π)s/^2 Γ((s+δ)/ 2 )L(s,χ),

whereδ=0 or 1 according asχ(− 1 )=1or−1, then

Λ( 1 −s,χ) ̄ =εχΛ(s,χ),

where

εχ=i−δm−^1 /^2

∑m

k= 1

χ( ̄ k)e^2 πik/m.

It follows from the functional equation that|εχ|=1. Indeed, by taking complex
conjugates we obtain, for reals,

Λ( 1 −s,χ)= ̄εχΛ(s,χ) ̄

and hence, on replacingsby 1−s,

Λ(s,χ)= ̄εχΛ( 1 −s,χ) ̄ =εχε ̄χΛ(s,χ).

The extended Riemann hypothesis implies that no DirichletL-functionL(s,χ)has
a zero in the half-planeRs> 1 /2, sincef(s)=

∏

χL(s,χ)is the Dedekind zeta-
function of the algebraic number fieldK=Q(e^2 πi/m). Hence it may be shown that if
the extended Riemann hypothesis holds, then

ψ(x;m,a)=x/φ(m)+O(x^1 /^2 log^2 x)

and

π(x;m,a)=Li(x)/φ(m)+O(x^1 /^2 logx),

where the constants implied by the O-symbols are independent of m and a.
Assuming the extended Riemann hypothesis, Bach and Sorenson (1996) have shown
that, for anya,mwith 1≤a<mand(a,m)=1, the least primep≡amodm
satisfiesp< 2 (mlogm)^2.
Without any hypothesis, Linnik (1944) proved that there exists an absolute
constantLsuch that the least prime in any arithmetic progressiona,a+m,a+ 2 m,...,
where 1≤a<mand(a,m)=1, does not exceedmLifmis sufficiently large.
Heath-Brown (1992) has shown that one can take anyL> 11 /2.