Dirichlet's Theorem on Arithmetic Progressions
Look at the primes ending in 1: 11, 31, 41, 61, 71, \dots
— they keep coming. What about primes of the form 4k + 3, or
100k + 7? Dirichlet's theorem answers all such questions
at once, and its proof launched the entire field of analytic number theory.
The statement
If a and m are
coprime,
then the arithmetic progression
a,\ a+m,\ a+2m,\ a+3m,\ \dots
contains infinitely many primes.
The coprimality condition is clearly necessary — if \gcd(a, m) = d > 1,
every term is divisible by d and at most one can be prime. Dirichlet
proved that this obvious obstruction is the only one. (Even more is true: the primes spread
themselves equally among the \varphi(m) valid residue classes.)
The idea of the proof
Dirichlet generalised the
Euler product proof
that there are infinitely many primes. He attached to each residue class a
Dirichlet series
twisted by a character \chi — a multiplicative function
on residues — forming an L-function:
L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}} = \prod_{p} \frac{1}{1 - \chi(p) p^{-s}}.
The characters act as filters that isolate a single progression. The crux — the hard technical
heart — is showing L(1, \chi) \ne 0 for every non-trivial character; that
non-vanishing is what guarantees the primes in the class never run dry.
Its significance
Dirichlet's theorem was the first triumph of applying continuous analysis to a discrete,
purely arithmetic question — the founding act of analytic number theory. The
L-functions it introduced are now central objects across mathematics,
and their own (conjectured) zero-free regions form a
generalised Riemann Hypothesis.