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I feel that this is a very stupid question to be asking, but I can't figure it out. I've been trying to figure out the Möbius inversion formula, with pretty much no experience in this direction at all, and ran into the following problem:

So according to the Möbius inversion formula, $$g(n)=\sum_{d|n}f(d) \Longleftrightarrow f(n)=\sum_{d|n}g\left(\frac nd\right)\mu(d)$$. Yes? Since this is an equivalence, it should work both ways. Let's say $g(n)=1$, in that case, plug it into the right-hand formula and find that $f(n)=0$, or at least Wikipedia tells me that $\sum_{d|n}\mu(d)=0$. Problem: when we look at the formula on the left, we see that $g(n)$ is now equal to the sum of a finite number of zeros, even though we defined it as $1$. What stupid mistake am I making here?

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$\sum_{d\mid n} \mu(d) = 0$ for $n > 1$. But $\sum_{d\mid 1} \mu(d) = 1$. – Daniel Fischer Jul 16 '14 at 21:37
Yes, but $n$ is not necessarily $1$, I think; it's the definition of $f(n)$, for $n \in \mathbb N$, so why should $n$ be restricted to $1$? – Laertes Jul 16 '14 at 21:39
But $$f(1) = \sum_{d\mid 1} g\left(\frac{1}{d}\right)\mu(d) = g(1)\mu(1) = g(1) = 1.$$ For $n > 1$, you have $f(n) = 0$. – Daniel Fischer Jul 16 '14 at 21:40
$f(n)$ is a function, it changes depending on $n$. So when $n=1$ you get $1$, only otherwise do you get $0$. – Adam Hughes Jul 16 '14 at 21:40
But $g(n)$ is supposed to be $1$ for all $n$. – Laertes Jul 16 '14 at 21:43
up vote 2 down vote accepted

The point is that

$$\sum_{d\mid n} \mu(d) = \begin{cases} 1 &, n = 1\\ 0 &, n > 1. \end{cases}$$

Thus, for $g \equiv 1$ you have

$$f(n) = \sum_{d\mid n} g\left(\frac{n}{d}\right)\mu(d) = \sum_{d\mid n} \mu(d) = \begin{cases} 1 &, n = 1\\ 0 &, n > 1 \end{cases}$$

and hence

$$g(n) = \sum_{d\mid n} f(d) = f(1) + \sum_{\substack{d\mid n\\ d > 1}} f(d) = f(1) + \sum_{\substack{d\mid n\\ d > 1}} 0 = f(1) = 1,$$

as it should be.

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