The coherent and incoherent channels of the neutrinoless muon to electron conversion in nuclei, µ − (A, Z) → e − (A, Z) * , are studied throughout the periodic table. The relevant nuclear matrix elements are computed by explicitly constructing all possible final nuclear states in the context of the quasi-particle RPA. The obtained results are discussed in view of the existing at PSI and TRIUMF experimental data for 48 T i and 208 P b and compared with results obtained by: (i) shell model sum-rule techniques (ii) nuclear matter mapped into nuclei via a local density approximation and (iii) earlier similar calculations.PACS number(s): 23.40.Bw, 13.35.Bv, 21.60.Jz, 12.60.Cn