Talk:Möbius inversion formula

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In the definition section, the article says

The formula is also correct if f and g are functions from the positive integers into some abelian group.

How is one to make sense of g(d) μ(n/d) if these values only have a group operation (addition) defined on them?

It seems that it should say "... into some commutative ring."

132.206.150.179 03:18, 10 December 2006 (UTC)

No. The function g takes values in the group, but the values of μ are plain integers, and every Abelian group is a Z-module. In other words, the meaning is
g(d)\mu(n/d)=\begin{cases}g(d),&\text{if }\mu(n/d)=1,\\-g(d),&\text{if }\mu(n/d)=-1,\\0,&\text{if }\mu(n/d)=0.\end{cases}
I reordered the formula to make it look like left module multiplication, which is hopefully more clear. -- EJ 12:40, 19 December 2006 (UTC)

I think the Moebius inversion formula was first introduced by Dirichlet and some other mathematician (perhaps Liouville). We should check the historical information ! —Preceding unsigned comment added by 132.206.33.23 (talk) 18:31, 28 February 2008 (UTC)