avoid a confusion, the distinction between $\mu^{\prime}$ for three dimensions and $\mu^{\prime 1D}$ is needed.)

The vacuum contribution to the 1D-thermodynamical potential after regularization is (see Appendix B)

$$\label{Grand-Potential-2Dvacreg}
\Omega^{1D}_{vac}(B)= -\frac{m^2}{2 \pi} ((1-b) \log (1-b)+ (1+b)\log (1+b)),$$

and finally for the 1D-thermodynamical potential in the limit of low temperature we have

$$\label{Grand-Potential-2DTcero}
\Omega^{1D}(B,\mu,T) =  - \frac{\sqrt{\varepsilon(0,B)}}{\sqrt{2 \pi} \beta^{3/2}} Li_{3/2}(e^{\mu^{\prime 1D} \beta}) -\frac{m^2}{2 \pi} \left ( (1-b) \log (1-b)+ (1+b)\log (1+b) \right ).$$

The particle density obtained from Eq. (\[Grand-Potential-2DTcero\]) is

$$\label{Particle-Density-2DTcero}
N^{1D} = -\frac{\partial \Omega^{1D}(B,\mu,T)}{\partial \mu}= \frac{\sqrt{\varepsilon(0,B)}}{\sqrt{2 \pi \beta^{1/2}}} Li_{1/2}(e^{\mu^{\prime 1D}  \beta}).$$

This expression is similar to the one obtained for a three dimensional charged vector boson gas in a constant magnetic field (see Eq. (23) of [@Khalilov1997]). It does not admit the substitution $\mu^{\prime 1D} = 0$ (because $Li_{1/2}(1)\rightarrow\infty$), therefore, a critical temperature can not be defined and the gas does not have BEC in the usual sense. Nevertheless, in the very low temperature limit, in which $\mu\sim \varepsilon(0,B)\gg T$ but $T \gg \mu^{\prime 1D}$ expression (\[Particle-Density-2DTcero\]) can be approximated as

$$\label{Particle-Density-2Dcondensate}
N^{1D} \simeq \frac{1}{2 \beta} \sqrt{\frac{2 \varepsilon(0,B)}{-\mu^{\prime 1D}}}.$$

Again, an expression equivalent to Eq. (\[Particle-Density-2Dcondensate\]) for a charged vector bosons gas has been already obtained in Ref. [@ROJAS1996148]. 