#### Date of Completion

Spring 5-1-2013

#### Thesis Advisor(s)

Keith Conrad

#### Honors Major

Mathematics

#### Disciplines

Mathematics

#### Abstract

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 + 3*k*, 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.

#### Recommended Citation

Stanford, Nicholas, "Dirichlet's Theorem and Applications" (2013). *Honors Scholar Theses*. 286.

https://opencommons.uconn.edu/srhonors_theses/286