fingerprint of the BEC phase transition. The maximum decreases and smoothes with the increment of the magnetic field, and is expected to disappear for $B \rightarrow B_c$ ($b \rightarrow 1$), because when $B=B_c$ the gas is condensed at any temperature and density.

![\[fig2\] The specific heat as a function of the temperature for $N=10^{32} cm^{-3}$ and several values of the magnetic field.](C.eps){width="0.5\linewidth"}

At this point, it is worthwhile to comment on the reasons for the difference in the nature of the BEC showed by the charged and the neutral vector boson gas in a constant magnetic field. As it was already known, the BEC in the charged gas is diffuse [@ROJAS1996148; @PEREZROJAS2000], while we just found it is usual for the neutral gas. The difference arises from the reduction in the dimension suffered by the charged gas for lows temepratures or strong fields. This reduction is a consequence of the quantized spectrum, since the perpendicular momentum component is replaced by Landau levels due to the coupling between the charge and the magnetic field. In the low temperature/high field limit the charged bosons concentrate in the lowest Landau Level (LLL) i.e, $n=0$ (which implies $p_{\perp}=0$), and therefore the three dimensional charged 