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A theory of the static electron polarizability of crystals whose energyspectrum is modified by quantizing magnetic fields is presented. It is arguedthat The polarizability is strongly affected by non-dissipative Hall currentsinduced by the presence of crossed electric and magnetic fields: these can evenchange its sign. Results are illustrated in detail for a two dimensional squarelattice. The polarizability and the Hall conductivity are respectively linkedto the two topological quantum numbers entering the so--called Diophantineequation. These numbers could in principle be detected in actual experiments.

Electron polarizability of crystalline solids in quantizing magnetic fields and topological gap numbers