Cooper instability in uniform electron gas: Irrelevance of Kohn-Luttinger mechanism

We study the Cooper instability in jellium model in the controlled regime of

 small to intermediate values of the Coulomb parameter r_s <= 2.  

 We confirm that superconductivity naturally emerges from purely repulsive interactions

 described by the Kukkonen-Overhauser vertex function. By employing

 the implicit renormalization approach we reveal that even in the small

 r_s limit, the dominant mechanism behind Cooper instability is based on

 dynamic screening of the Coulomb interaction, whereas the Kohn-Luttinger

 contribution is negligibly small and, thus, not relevant.