  The thermodynamic properties of electron gas under the extreme conditions of
high temperature, high matter density, and/or a strong magnetic field largely
determine the behaviour of matter in upper layers of neutron stars and
accretion columns of magnetized neutron stars in binary systems. A strong
magnetic field in these objects makes the motion of electrons across the field
essentially quantum. The possible electron degeneracy and relativism of
electrons are also important. When studying accretion onto a magnetar in a
binary system, the intensive generation of electron-positron pairs in the
quantizing magnetic field should also be taken into account. We consider in
detail the thermodynamic properties of a gas of free electrons in strong
magnetic fields, taking into account their relativism and degeneracy, as well
as the equilibrium creation of electron-positron pairs in a high-temperature
plasma in the presence of a quantizing magnetic field.
