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Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.
The proof of Dirichlet's theorem suggests a method for defining a notion of density of a set of primes, called its Dirichlet density, and the primes of the form a+mk have a Dirichlet density 1/φ(m), which is independent of a. While the definition of Dirichlet density is not intuitive, it is easier to compute than a more natural concept of density, and the two notions of density turn out to be equal when they both exist.
Dirichlet's theorem is often used to show a prime number exists satisfying a particular congruence condition while avoiding a finite set of "bad" primes. For example, it allows us to find the density of the set of primes p such that a given nonzero integer a is or is not a square mod p. More generally, it lets us find the density of the set of primes at which a finite set of integers have prescribed Legendre symbol values.
Stanford, Nicholas, "Dirichlet's Theorem and Applications" (2013). Honors Scholar Theses. 286.