Möbius inversion across all natural numbers (no divisors)

by Richard Burke-Ward   Last Updated November 10, 2018 11:20 AM

I'm trying (self-taught) to understand more about Möbius inversion. Take two arithmetic functions $f$ and $g$ defined by

$$g(n)=\sum_{d|n}f(d)$$

(Presumably $d$ refers to divisors $>0$ and $\le n$). As I understand it, the inverse via Möbius inversion is then

$$f(n)=\sum_{d|n}\mu(d)g\biggl(\frac{n}{d}\biggr)$$

My question is this: is it possible to modify this to invert sums that run across all natural numbers? In other words, can I invert the following function?

$$g(n)=\sum_{n=1}^\infty f(n)$$

Presumably I need to find a bijective relationship between $d$ and $\mathbb{N}$. If so, how? Or is there a better approach?



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