Can we define Mobius function for any real number and any complex number ?

Question

All:

To me, Mobius function is a bit mysterious. I just want to know if we can define Mobius function for any real number or any complex number ?

Can anyone point out any resource on this ?

Thank you.

Answer 1

I don't know what you mean by defining it for any real or complex number.

But (assuming that by Möbius function you mean the standard thing whose value at square-free numbers is $\pm1$ depending on whether the number of prime factors is even or odd, and equal to $0$ elsewhere) there is a thing called an incidence algebra.

Every locally finite partially ordered set has an incidence algebra, in which the identity element for multiplication is called the $\delta$ function, the function everywhere equal to $1$ is the $\zeta$ function, and the inverse of the $\zeta$ function is the $\mu$ function. In the special case in which the poset is the positive integers ordered by divisibility, the $\mu$ function is the standard $\mu$ function mentioned in the paragraph above.

PS: I just realized that what you meant was: "Can one define $\mu(x)$ for $x\in\mathbb R$ or $x\in\mathbb C$?". I would guess not in any reasonable way.