shall note $\omega\equiv\omega_{1}=\omega_{-1}=\mu^{2}/\alpha^{2}$ and $\Delta\equiv-\Delta_{1}=\Delta_{-1}=2\mu_{\mathrm{B}}/\mu$. Then we have

$$M_{SO}=\sum_{s=\pm1}A_{s}\sum_{p=1}^{\infty}\frac{1}{\pi p}\sin\left[2\pi p\omega\left(\frac{1}{B}+s\Delta\right)\right].\label{M sawtooth}$$

Equation (\[M sawtooth\]) gives the SO contribution to the MO. There is still another oscillatory contribution, which comes from the continuous oscillation in $\Omega_{C}$. This oscillation is not a SO, but an expression for it can be straightforward obtained noticing that $M_{osc}=-\mathcal{A}^{-1}\left(\partial\Omega_{C}^{osc}/\partial B\right)_{\mu}$. From equation (\[M sawtooth\]) we get that $\Omega_{C}^{osc}$ should be of the form $\Omega_{C}^{osc}=\sum_{s=\pm1}C_{s}\sum_{p=1}^{\infty}\left(\pi p\right)^{-2}\cos\left[2\pi p\omega\left(\frac{1}{B}+s\Delta\right)\right]$, where $C_{s}$ satisfies $M_{osc}=-\mathcal{A}^{-1}\left(\partial\Omega_{C}^{osc}/\partial B\right)_{\mu}$, so $2\omega C_{s}/\mathcal{A}B^{2}=-A_{s}$. Therefore the MO is given by

$$\begin{aligned}
\fl M_{osc} = \sum_{s=\pm1}\left\{ \left[\frac{B}{\omega\phi}\left(-\mu+\frac{3}{2}s\mu_{\mathrm{B}}B\right)\right]\sum_{p=1}^{\infty}\frac{1}{\left(\pi p\right)^{2}}\cos\left[2\pi p\omega\left(\frac{1}{B}+s\Delta\right)\right]\right.\nonumber\\
\left.+A_{s}\sum_{p=1}^{\infty}\frac{1}{\pi p}\sin\left[2\pi p\omega\left(\frac{1}{B}+s\Delta\right)\right]\right\}.\label{Mosc}\end{aligned}$$

Equation (\[Mosc\]) is in agreement with the result found in [@Sharapov2004], where the oscillating part of the magnetization is expressed as an infinite series. To obtain the total magnetization we also have to add the non-oscillatory contribution from both the valence and conduction band. Nevertheless, it can be shown [@Sharapov2004] that in graphene, at $\mu>0$, this contribution cancels and the total magnetization is just given by equation (\[Mosc\]). This result can be intuitively understood by noticing that the non-oscillatory contribution to $M$ comes from the variation of the degeneracy and energy as $B$ is changed (which gives a variation proportional to $\sqrt{B}$, for $D\varepsilon\sim 