The pressure of QCD at finite temperature and quark number density

This paper is a slightly modified version of the introductory part of a doctoral dissertation that contained also three original articles, hep-ph/0212283, hep-ph/0305183 and hep-ph/0311323. Our purpose is to review the history and present status of finite-temperature perturbation theory as applied to the context of determining the equilibrium properties of quark-gluon plasma, most notably the pressure of QCD at finite temperatures and quark chemical potentials. We first introduce the general formalism of finite-temperature field theory and perturbation theory, then follow through the evaluation of the pressure order by order, and finally proceed to analyze the most recent, order g6ln g results by comparing the perturbative predictions with lattice data. In the appendix we provide a somewhat pedagogical introduction to the most important computational techniques used in the perturbative framework, namely the analytic evaluation of multi-loop vacuum diagrams both in full QCD and in its three-dimensional high-T effective theories.