Primes in Arithmetic Progressions

Specialeforsvar: Mikkel Andersen Drygaard

Titel: Primes in Arithmetic Progressions

Abstract: In this thesis we investigate how primes behave in arithmetic progressions. We shall see that there are, in fact, infinitely many primes p ≡ a (mod q) for each coprime a, q ∈ N. In particular, we show the prime number theorem for arithmetic progressions, which is the main interest of the thesis. The starting point is to define Dirichlet characters and establish some basic properties concerning them. This is then used to define the Dirichlet L-functions, L(s, χ). We then show that L(1 + iτ, χ) ̸= 0 for χ ̸= χ0 and that this yields an initial proof of the prime number theorem for arithmetic progressions, with a rather weak error term. To obtain better error terms, a finer study of the analytic properties of the L-functions is necessary. We shall see that L(s, χ) has an meromorphic continuation to all of C, by proving a functional equation for L-functions, which is then used to investigate zeroes of the L-functions. We even obtain a zero-free region in the critical strip, except for the potential Siegel zero. In the final chapter, we use our zero-free region to prove the Siegel-Walfisz theorem, which provides a substantial improvement to the error term. Lastly, we briefly discuss potential improvements and limitations to the error term.

Vejleder: Jasmin Matz
Censor:  Simon Kristensen, Aarhus Universitet