This adventure starts with the well known formulae

during a time I was fond to invert a lot of formulae using the Moebius function. The substitution of by allows to obtain

And the substitution of by gives

And my adventure follows in this way

So, using the fact that we have

and

At this moment I thougt it could be interesting to evaluate the double sum but changing the order of the summatories. Once more I am going to take into account that .

It is well known that in case that .So we have

And the conmutator

is equal to

or to

This adventure would be very interesting if the value of the conmutator were zero. In fact I was temted to think so.

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