This adventure starts considering the known identity

I inverted it using Mobius´ function. For simplicity I called the above second member.

The substitution of by gives

I had the idea of making another sum, like for instance

Some easy manipulation allow us to obtain

Lets look at the result

It is interesting to observe that

And so

The properties of the Moebius´ function allow us to write

This adventure would be very interesting if we could conmute the two sumatories of the first member.

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