A detailed calculation of the real part of the finite temperature dynamic s
usceptibility of the free Bose gas is presented. After a short discussion o
n the different ways in which it can he calculated for temperatures above a
nd below the Bose-Einstein transition temperature, its main properties and
its evolution with q and T are analyzed. Finally, expressions for the lowes
t-order energy weighted sum rules are also derived and studied. (C) 2001 Pu
blished by Elsevier Science B.V.