Mobius‘ function has the important property of inverting functional relations [1] of the type and giving in turn (Also relations (0.1) and (0.2) follow respectively from (0.11) and (0.21))

I establish in this paper that no other arithmetical function μ*(n) can perform the above inversion. In other words, I prove that Inversion is the characteristic property of Mobius‘ μ-function. Theorem 1 below asserts that if for one pair of functions g(n) and f(n) (subject to a special condition) relations (0.1) and (0.11) are true with an arithmetical function μ*(n) (possibly depending on g and f), then μ*(n) coincides with the Mobius‘-function. Theorem 2 below gives a similar result in connection with functions G(x) and F(x) defined for all real x ≥ 1.