  We discuss the sign of the persistent current of N electrons in one
dimensional rings. Using a topology argument, we establish lower bounds for the
free energy in the presence of arbitrary electron-electron interactions and
external potentials. Those bounds are the counterparts of upper bounds derived
by Leggett. Rings with odd (even) numbers of polarized electrons are always
diamagnetic (paramagnetic). We show that unpolarized electrons with N being a
multiple of four exhibit either paramagnetic behavior or a superconductor-like
current-phase relation.
