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Precision SU(3) lattice thermodynamics for a large temperature range

We present the equation of state (pressure, trace anomaly, energy density and entropy density) of the SU(3) gauge theory from lattice field theory in an unprecedented precision and temperature range. We control both finite size and cut-off effects. The studied temperature window (0.7...1000 T_c) stretches from the glueball dominated system into the perturbative regime, which allows us to discuss the range of validity of these approaches. We also determine the preferred renormalization scale of the Hard Thermal Loop scheme and we fit the unknown g^6 order perturbative coefficient at extreme high temperatures T>100 T_c. We furthermore quantify the nonperturbative contribution to the trace anomaly using a simple functional form. Our high precision data allows one to have a complete theoretical description of the equation of state from T=0 all the way to the phase transition, through the transition region into the perturbative regime up to the Stefan-Boltzmann limit. We will discuss this description, too.