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

.