Analytic Number Theory and Dirichlet’s Theorem

Dirichlet’s Theorem on Primes in an Arithmetic Progression

Our goal is to prove the following theorem: Dirichlet’s Theorem: For any coprime a, b ∈ Z, there are infinitely many primes p such that p ≡ a (mod b). Although the statement of the theorem involves only integers, the simplest proof requires the use of complex numbers and Dirichlet L-series. Most of this paper will therefore be devoted to proving some basic properties of characters and L-series,...

The Dedekind Zeta Function and the Class Number Formula Math 254B Final Paper

The aim of this paper is to prove that the Dedekind zeta function for a number field has a meromorphic continuation to the complex plane, obtaining the analytic class number formula with it, and to present some of the applications of these results. In particular, It will be shown its use in proving Dirichlet’s prime number theorem and in calculating the class number of quadratic fields. The pro...

Dirichlet’s Theorem on Primes in Arithmetic Progressions

Let us be honest that the proof of Dirichlet’s theorem is of a difficulty beyond that of anything else we have attempted in this course. On the algebraic side, it requires the theory of characters on the finite abelian groups U(N) = (Z/NZ)×. From the perspective of the 21st century mathematics undergraduate with a background in abstract algebra, these are not particularly deep waters. More seri...

Two classic theorems from number theory: The Prime Number Theorem and Dirichlet’s Theorem

Dirichlet Prime Number Theorem

In number theory, the prime number theory describes the asymptotic distribution of prime numbers. We all know that there are infinitely many primes,but how are they distributed? Dirichlet’s theorem states that for any two positive coprime integers a and d, there are infinitely many primes which are congruent to a modulo d. A stronger form of Dirichlet’s theorem states that the sum of the recipr...