-\frac{\partial \Omega_{st}}{\partial B}-\frac{\partial \Omega_{vac}}{\partial B},$$

$$\label{magnetization-1D}
M^{1D}=-\frac{\partial \Omega^{1D}_{st}}{\partial B}-\frac{\partial \Omega^{1D}_{vac}}{\partial B}.$$

In Eq. (\[magnetization\]) $N_0$ is the number of particles in the condensate and $d$ is the effective magnetic moment Eq.(\[magmoment\]). This term has to be added because in the low temperature limit all the bosons are aligned to the field and contributes to the magnetization, but for temperatures under $T_{cond}$, $\Omega_{st}$ only accounts for the particles that are out of the condensate.

For the statistical contributions to the magnetization we have the expressions

$$\label{magnetization-st}
M_{st}=d N_0 -\frac{\partial \Omega_{st}}{\partial B}= \frac{\kappa m}{\varepsilon(0,B)} N - \frac{2 \kappa m T^{5/2}}{(4 \pi)^{5/2} (2-b)^2 \varepsilon(0,B)^{1/2}} Li_{5/2}(e^{\beta \mu^{\prime}}),$$

$$\label{magnetization-1Dst}
M^{1D}_{st}=-\frac{\partial \Omega^{1D}_{st}}{\partial B}= \frac{\kappa m}{\varepsilon(0,B)} N - \frac{\kappa m T^{3/2} }{2^{3/2} \pi^{1/2} \varepsilon(0,B)^{3/2}} Li_{3/2}(e^{\beta \mu^{\prime}}),$$

while the vacuum contributions are

$$\label{magnetization-vac}
M_{vac}= -\frac{\kappa m^3}{72 \pi} \left( 7 b(b^2-6) + 3(2 b^2+2 b-7)(1-b)\log(1-b)-3(2b^2-2b-7)(1+b)\log(1+b)\right ),$$

$$\label{magnetization-1Dvac}
M^{1D}_{vac}= \frac{\kappa m}{\pi} \log \left(\frac{1+b}{1-b}\right ).$$

It is possible to show that for $T\ll m$ the second terms in Eq.(\[magnetization-st\]) and Eq.(\[magnetization-1Dst\]) are negligible, as well as one can prove that the vacuum magnetization (Eqs. (\[magnetization-vac\]) and (\[magnetization-1Dvac\])) is only relevant for low particle densities at very high fields, so it can be also neglected. Finally, for both, one and three dimensions, 