The main part of this thesis deals with the zero-field diamagnetic susceptibility of a Bloch electrons gas at fixed temperature and fixed density in the limit of low temperatures. For a free electrons gas (that is when the periodic potential is zero), the steady diamagnetic susceptibility has been computed by L. Landau in 1930; the result is known as Landau formula. As for the Bloch electrons, E.R. Peierls in 1933 showed that under the tight-binding approximation, the formula for the diamagnetic susceptibility remains the same but with the mass of the electron replaced by its ''effective mass''; this result is known as the Landau-Peierls formula. Since, there were very many attempts in order to clarify the assumptions of validity of the Landau-Peierls formula. The main result of this thesis establishes rigorously that at zero temperature, as the density of electrons tends to zero, the leading contribution of the diamagnetic susceptibility is given by the Landau-Peierls formula with the effective mass of the lowest Bloch energy band.
