  We study the influence of inelastic electron-electron scattering on the
temperature variation of the Seebeck coefficient in the normal phase of
quasi-one-dimensional organic superconductors. The theory is based on the
numerical solution of the semi-classical Boltzmann equation for which the
collision integral equation is solved with the aid of the electronic umklapp
scattering vertex calculated by the renormalization group method. We show that
the one-loop renormalization group flow of momentum and temperature dependent
umklapp scattering, in the presence of nesting alterations of the Fermi
surface, introduce electron-hole asymmetry in the energy dependence of the
anisotropic scattering time. This is responsible for the enhancement of the
Seebeck coefficient with respect to the band $T$-linear prediction and even its
sign reversal around the quantum critical point of the phase diagram, namely
where the interplay between antiferromagnetism and superconductivity along with
the strength of spin fluctuations are the strongest. Comparison of the results
with available data on low dimensional organic superconductors is presented and
critically discussed.
