The Mobius inversion formula and inverting lattice sums

The Mobius inversion formula is an interesting theorem from number theory that has application to a number inverse problems, particularly inverse lattice problems. Specific inverse problems, however, often require related Mobius inversion formulae that can be derived from the fundamental formula. Derivation of such formulae is not easy for the non-specialist, however. Examples of the kinds of inversion formulae that can be derived and their application to inverse lattice problems are described.