Mobius Function: prove function is multiplicative

[ifoan]

I'm back again :\

I need to prove that the Mobius Function is multiplicative.

I know that an arithmetical function \(\displaystyle \mu\) is called multiplicative if:

\(\displaystyle \mu(mn)=\mu(m)\mu(n)\)whenever \(\displaystyle gcd(m,n) = 1.\)

So, I can right away see that it IS multiplicative because gcd(1,5)=1.
Thus:

\(\displaystyle \mu(5)=\mu(5)\mu(1)\)

\(\displaystyle \mu(5)=-1\)
\(\displaystyle \mu(1)=1\)

So it follows:

-1=-1

Showing that:
\(\displaystyle \mu(mn)=\mu(m)\mu(n)\)

This is what I have so far, but my professor doesn't like for us to use concrete examples. This is concrete, I believe. So how else can I prove this?

 

[pka]

It is often very difficult for students to understand but this is true: There are no standard or uniform definitions in mathematics. The problems that you have posted seem to be unique to the text or course you are following. Without access to the definitions, it is almost impossible to help. You need to include the definitions and any particulars about the problem.

I can give you a website that may be what you are working with:
http://mathworld.wolfram.com/MoebiusFunction.html.
About halfway down that page is a mention of the ‘multiplicative property’ (16).

 

[ifoan]

Thanks pka

I guess im going to try my best and help for the best