It is now a truism that apparent defects in our methods of studying and teaching mathematics are causing anxiety to all those who are concerned with the subject. There is, in fact, a widespread feeling that some positive action is needed—together with some uncertainty about what the action should be and who should do it. The inevitable feeling of frustration that follows is hardly surprising in the light of the technical complexity of the subject and the diffusion of control among the several bodies which are responsible for it.

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.

The formation of a Joint Mathematical Council for the United Kingdom has recently been announced, and readers of the Gazette may be interested to know the reasons for setting up such a body, and what it hopes to achieve.