  This paper proves the mean field limit and quantitative estimates for
many-particle systems with singular attractive interactions between particles.
As an important example, a full rigorous derivation (with quantitative
estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is
obtained for the first time. To give an answer to this longstanding problem, we
take advantage of a new modulated free energy and we prove some precise large
deviation estimates encoding the competition between diffusion and attraction.
Combined with the range of repulsive kernels, already treated in the
s{\'e}minaire Laurent Schwartz proceeding
[https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the full
proof of results announced by the authors in [C. R. Acad. Sciences Section
Maths (2019)].
