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\begin{document}
\begin{equation}
\lim _{q\rightarrow 0}\frac{1}{L^{d}}\sum_{ \vec{k}}\frac{n_{\vec{k}\sigma}-n_{\vec{k}+\vec{q}\sigma}}{\varepsilon _{\vec{k}\sigma}-\varepsilon_{\vec{k}+\vec{q}\sigma}}=\frac{1}{L^{d}}\sum _{\vec{k}}\frac{\partial n_{\vec{k}\sigma}}{\partial \varepsilon _{\vec{k}\sigma}}{\ndtstile{}{T=0}}-\frac{1}{L^{d}}\sum _ { \vec{k}}\delta (\epsilon _{F\sigma}-\varepsilon _{\vec{k}\sigma})=-N_{\sigma}(0)
\end{equation}    
 \end{document}