A combinatorial method to introduce Möbius inversion formula and Möbius function (Q1801763)

The cohesive energy \(E(x)\) for each atom in an infinite linear chain can be expressed as a sum \(E(x)=\sum^ \infty_{n=1}\varphi(nx)\), \(\varphi\) being the pairwise potential. This equation allows the inversion \(\varphi(x)=\sum^ \infty_{n=1}I(n)E(nx)\), where the inversion coefficient \(I(n)\) is just the Möbius function \(\mu(n)\) from number theory. Both equations are thus the Möbius transform, which says, that \(F(x)=\sum^ \infty_{n=1}f(nx)\) if and only if \(f(x)=\sum^ \infty_{n=1}\mu(n)F(nx)\).