$$\label{Grand-Potential-2Dsst1}
\Omega^{1D}_{st}(B,\mu,T)= -\sum_{s=-1,0,1} \frac{1}{2 \pi \beta}  \sum_{n=1}^{\infty} \frac{e^{n \mu \beta}+e^{- n \mu \beta }}{n} \int\limits_{0}^{\infty} dp_3 e^{-n \beta \varepsilon(s)},$$

where, as for the three dimensional gas, the terms $e^{n \mu \beta}$ and $e^{- n \mu \beta}$ stand for the particles and the antiparticles respectively. The integration over $p_3$ can be carried out, and we obtain for $\Omega^{1D}_{st}(B,\mu,T)$ the expression

$$\label{Grand-Potential-2Dsst3}
\Omega^{1D}_{st}(B,\mu,T)= -\sum_{s=-1,0,1} \frac{m \sqrt{1- s b}}{ \pi \beta}  \sum_{n=1}^{\infty} \frac{e^{n \mu \beta}+e^{- n \mu \beta }}{n} K_1(n \beta m \sqrt{1-s b}).$$

$K_1(x)$ is the first order McDonald function.

Performing the sum over the spin we have

$$\begin{aligned}
\label{Grand-Potential-2Dstexact}
\Omega^{1D}_{st}(B,\mu,T)= - \frac{m \sqrt{1+ b}}{ \pi \beta}  \sum_{n=1}^{\infty} \frac{e^{n \mu \beta}+e^{- n \mu \beta }}{n} K_1(n \beta m \sqrt{1+b})\\ \nonumber
-\frac{m}{ \pi \beta}  \sum_{n=0}^{\infty} \frac{e^{n \mu \beta}+e^{- n \mu \beta }}{n} K_1(n \beta m)\\ \nonumber
-\frac{m \sqrt{1-b}}{ \pi \beta}  \sum_{n=0}^{\infty} \frac{e^{n \mu \beta}+e^{- n \mu \beta }}{n} K_1(n \beta m \sqrt{1-b}). \nonumber\end{aligned}$$

As for the three dimensional gas, in the low temperature limit the leading term of Eq. (\[Grand-Potential-2Dstexact\]) comes from the particles with $s=1$. This leading term can be re-written as

$$\label{Grand-Potential-2DstTcero}
\Omega^{1D}(B,\mu,T)= - \frac{\sqrt{\varepsilon(0,B)}}{\sqrt{2 \pi} \beta^{3/2}} Li_{3/2}(e^{\mu^{\prime 1D} \beta}),$$

with $\mu^{\prime 1D} =\mu-\varepsilon(0,B)$. (Besides they have the same definition, to 