El teorema dels nombres primers i el teorema de Lee-Yang

Abstract

The first objective of this work it to present a detailed study on the Riemann $\zeta$ function as a complex function, which includes its analytic continuation, functional equation and those necessary properties to proof in the following chapter the prime number theorem. The second objective is to write a self contained proof of the primer number theorem, discarding the superfluous results found in the references to achieve a clear and concise proof. In that regard, we also include our own graphics to help the understanding of the behavior of the functions that appear in the proof. Lastly, we consider the Riemann hypothesis, qualified as one of the Millennium Problems by the Clay Institute and a key point in the development of modern number theory. The second part of this work aims to find physical applications of the mathematical methods used in this work of analytic number theory. On this subject we present the Casimir effect, which is a first example on renormalization in quantum theories and in which we show that the analytic continuation of the $\zeta$ function plays an important role. The second case is Lee-Yang theory. Although in this last case there is no direct application of the methods studied in the mathematical part of the work, it shows the importance of studying the distribution of zeros of certain functions. Therefore, Lee-Yang theory could be a very interesting bridge between two apparently very different disciplines such as statistical mechanics and number theory. In fact, in the development of this work, we unexpectedly discovered that Lee and Yang based a part of their proof of the circle theorem presented here on a Polya article ([11]) on the integral representations of the Riemann $\zeta$ function, as Kac mentions in a comment of the same article. Lastly, we have applied the Lee-Yang theory to the one and two dimensional Ising models. We have been able to compute the distribution of the zeros of the partition functions, which has enabled us the study of the phase transitions of these thermodynamic systems.