Jens Nordström: Mönster bland primtalen: en introduktion till Riemanns zetafunktion

Abstract.

This thesis explores one of the most important results in the field of analytic number theory, namely the prime number theorem, which is a result regarding the distribution of the prime numbers. The thesis begins with an introduction to prime numbers and the historical study of these before we state the main result of this very inquiry, the prime number theorem. The main section of the thesis consists of a comprehensive proof of this theorem, and for this proof we first need to state a number of basic results about arithmetic functions and the Riemann zeta function. The thesis concludes with an introduction to the, to this day unsolved, problem of the Riemann hypothesis and its consequences for the distribution of the prime numbers.