1.3. L-functions 13
Figure 4. Johann Peter Gustav Lejeune Dirichlet (1805-
1859) and Georg Friedrich Bernhard Riemann (1826-1866).
where χ is a given Dirichlet character. For example, one can take χ0
to be the trivial Dirichlet character, i.e., χ0(n) = 1 for all n ≥ 1. Then
L(s, χ0) is the Riemann zeta function ζ(s). Dirichlet L-functions also
have Euler products:
L(s, χ) =
∞
n=1
χ(n)
ns
=
p
1
1 − χ(p)p−s
.
The idea of the proof of Dirichlet’s theorem generalizes the fol-
lowing proof, due to Euler, of the infinitude of the primes. Consider
ζ(s) =
∑∞
n=1
1
ns
=
p
1
1−p−s
and suppose there are only finitely many
primes. Then the product over all primes is finite, and therefore its
value at s = 1 would be finite (a rational number, in fact). However,
ζ(1) =
∑∞
n=1
1/n is the harmonic series, which diverges! Therefore,
there must be infinitely many prime numbers.
Dirichlet adapted this argument by looking instead at a different
function:
Ψa,N (s) =
p≡a mod N
1
ps
.
