Some remarks on the thermal and vacuum fluctuations of a massive scalar field

Thermal fluctuations of a massive scalar field in the Rindler wedge have been recently obtained. As a by product, the Minkowski vacuum fluctuations seen by a uniformly accelerated observer have been determined and confronted with the corresponding Minkowski thermal fluctuations of the same field, seen by an inertial observer. Since some of the calculations of this previous work have not been detailed on it, and they present some important subtleties, they are explicitly done here. These subtleties have to do with the leading order behaviour of certain parameter dependent integrals. Some of the leading order expansions are derived using the Riemann-Lebesgue lemma.