Josephson current through a Luttinger liquid.

We study the Josephson effect through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. The interactions are treated in the framework of the one-channel Luttinger model. At zero temperature, the Josephson critical current is found to decay algebraically with increasing distance between the junctions. The exponent is proportional to the strength of the Coulomb interaction. If the Luttinger liquid has a finite size, the Josephson current depends on the total number of electrons modulo 4. These parity effects are studied for the ring, coupled capacitively to a gate-voltage and threaded by a magnetic flux. The Josephson current changes continuously as a function of the gate voltage and {\em stepwise} as a function of the magnetic flux. The electron-electron interaction introduces {\em qualitatively} new features compared to the non-interacting case.