  An exhaustive investigation of metallic electronic transport and
superconductivity of organic superconductors (TMTSF)_2PF_6 and (TMTSF)_2ClO_4
in the Pressure-Temperature phase diagram between T=0 and 20 K and a
theoretical description based on the weak coupling renormalization group method
are reported. The analysis of the data reveals a high temperature domain
(T\approx 20 K) in which a regular T^2 electron-electron Umklapp scattering
obeys a Kadowaki-Woods law and a low temperature regime (T< 8 K) where the
resistivity is dominated by a linear-in temperature component. In both
compounds a correlated behavior exists between the linear transport and the
extra nuclear spin-lattice relaxation due to antiferromagnetic fluctuations. In
addition, a tight connection is clearly established between linear transport
and T_c. We propose a theoretical description of the anomalous resistivity
based on a weak coupling renormalization group determination of
electron-electron scattering rate. A linear resistivity is found and its origin
lies in antiferromagnetic correlations sustained by Cooper pairing via
constructive interference. The decay of the linear resistivity term under
pressure is correlated with the strength of antiferromagnetic spin correlations
and T_c, along with an unusual build-up of the Fermi liquid scattering. The
results capture the key features of the low temperature electrical transport in
the Bechgaard salts.
