I'm back again :\
I need to prove that the Mobius Function is multiplicative.
I know that an arithmetical function μ is called multiplicative if:
μ(mn)=μ(m)μ(n)whenever gcd(m,n)=1.
So, I can right away see that it IS multiplicative because gcd(1,5)=1.
Thus:
μ(5)=μ(5)μ(1)
μ(5)=−1
μ(1)=1
So it follows:
-1=-1
Showing that:
μ(mn)=μ(m)μ(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?
I need to prove that the Mobius Function is multiplicative.
I know that an arithmetical function μ is called multiplicative if:
μ(mn)=μ(m)μ(n)whenever gcd(m,n)=1.
So, I can right away see that it IS multiplicative because gcd(1,5)=1.
Thus:
μ(5)=μ(5)μ(1)
μ(5)=−1
μ(1)=1
So it follows:
-1=-1
Showing that:
μ(mn)=μ(m)μ(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?