We propose a theory of longitudinal resistivity in the normal phase of quasi-one-dimensional organic superconductors near the quantum critical point where antiferromagnetism borders with superconductivity under pressure. The linearized semi-classical Boltzmann equation is solved numerically, fed in by the half-filling electronic umklapp scattering vertex as derived from one-loop renormalization group calculations for the quasi-one-dimensional electron gas model. The momentum and temperature dependence of umklapp scattering has an important impact on the behaviour of longitudinal resistivity in the the normal phase. Resistivity is found to be linear in temperature around the quantum critical point at which spin-density-wave order joins superconductivity along the antinesting axis, to gradually evolve towards the Fermi liquid behaviour in the limit of weak superconductivity. A comparison is made between theory and experiments performed on the (TMTSF)$_2$PF$_6$ member of the Bechgaard salt series under pressure.
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