TheoremProved

Dirichlet's theorem on primes in arithmetic progressions

Statement

If $a$ and $n$ are coprime positive integers, the arithmetic progression $a,\ a+n,\ a+2n,\dots$ contains infinitely many primes.

Why is it true?

As long as a residue class modulo $n$ is not automatically ruled out (i.e. $\gcd(a,n)=1$ ), primes are spread evenly enough across residue classes that each admissible one gets infinitely many — no coprime residue class is somehow 'prime-free'.

Proof sketch

Introduce Dirichlet characters $\chi$ modulo $n$ and the associated L-functions $L(s,\chi)=\sum_k \chi(k)k^{-s}$ . Show $L(1,\chi)\ne0$ for every non-principal character $\chi$ . Combine, via orthogonality of characters, the divergence of $\sum_p p^{-s}$ (from the principal character) with the finiteness of $\log L(s,\chi)$ near $s=1$ for non-principal $\chi$ , to isolate the residue class $a$ and show $\sum_{p\equiv a\,(n)} 1/p$ diverges — forcing infinitely many such primes.

Stated by

Peter Gustav Lejeune Dirichlet

Topics that use this theorem

Prime number theorem

Related theorems

Euler product formula for ζ(s)Prime number theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Peter Gustav Lejeune Dirichlet (1837). Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält
Tom M. Apostol (1976). Introduction to Analytic Number Theory · DOI:10.1007/978-1-4757-5579-4