Prime Numbers

1.4 Analytic number theory 41
(mod D) is isomorphic to the multiplicative group Z∗ D of residues (mod D)
coprime to D. To conclude our brief tour of Dirichlet characters we record the
following two (dual) identities, which express a kind of orthogonality:
ϕ(D), if n ≡ 1(modD),
χ(n) =
(1.27)
0, if n ≡ 1(modD),
χ (mod D)
D
χ(n) =
n=1
ϕ(D), if χ is the principal character (mod D)
0, if χ is a nonprincipal character (mod D). (1.28)
Now we can turn to the main topic of this section, Dirichlet L-functions.
If χ is a Dirichlet character modulo D, let
L(s, χ) =
∞
n=1
χ(n)
.
ns The sum converges in the region Re(s) > 1, and if χ is nonprincipal, then
(1.28) implies that the domain of convergence is Re(s) > 0. In analogy to
(1.18) we have
L(s, χ) =
p
1 − χ(p)
ps −1 . (1.29)
It is easy to see from this formula that if χ = χ0 is the principal character
(mod D), then L(s, χ0) =ζ(s)
p|D (1 − p−s ), that is, L(s, χ0) isalmostthe
same as ζ(s).
Dirichlet used his L-functions to prove Theorem 1.1.5 on primes in a
residue class. The idea is to take the logarithm of (1.29) just as in (1.19),
getting
ln(L(s, χ)) = χ(p)
+ O(1), (1.30)
ps p
uniformly for Re(s) > 1 and all Dirichlet characters χ. Then, if a is an integer
coprime to D, wehave
χ (mod D)
χ(a)ln(L(s, χ)) =
χ (mod D)
= ϕ(D)
p
p≡a (mod D)
χ(a)χ(p)
p s
+ O(ϕ(D))
1
+ O(ϕ(D)), (1.31)
ps where the second equality follows from (1.27) and from the fact that
χ(a)χ(p) =χ(bp), where b is such that ba ≡ 1(modD). Equation (1.31) thus
contains the magic that is necessary to isolate the primes p in the residue class
a (mod D). If we can show the left side of (1.31) tends to infinity as s → 1 + ,
then it will follow that there are infinitely many primes p ≡ a (mod D), and
in fact, they have an infinite reciprocal sum. We already know that the term