In this scenario, electrons form pairs at a temperature "T*" that can be much larger than the critical temperature "T"c where superconductivity appears. Values of "T*" of the order of 300 K have been measured in underdoped cuprates where "T"c is about 80 K. The superconductivity does not appear at "T*" because large phase fluctuations of the pairing field cannot order at this temperature. The pseudogap is then produced by incoherent fluctuations of the pairing field. The pseudogap is a normal state precursor of the superconducting gap due to local, dynamic pairing correlations. This point of view is supported by a quantitative approach of the attractive pairing model to specific heat experiments.