Ok. I figured it out. I'll write my solution here so that it will be easy to find :).
ρ_{L}= \frac{9 π \hbar}{2e^{2}mNk_{b}T}\cdot \frac{1}{R^{2}σ^{2}}\int\int \frac{(K \cdot u)^{2}(K \cdot e)^{2}ζ(K)}{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}\frac{dσ}{\upsilon}\frac{dσ\;'}{\upsilon\;'} =...
In "Electrons and Phonons" by Ziman which you can find here, I have a problem with equation (9.5.14) on page 360 and (9.5.17) on page 361. I don't understand how from the first equation where you have double surface integral you obtain the second equation with only one integral over q.
I...