Möbius inversion formulas for flows of arithmetic semigroups (Q2469225)

Let \(S\) be an arithmetic semigroup, \(R\) a complete valued ring, \(M\) an \(R\)-module and let \(X\) be a non-empty set. A map \(\varphi: S\times X\rightarrow X\) is called an \(S\)-flow on \(X\) if \(\varphi(s,\varphi(t,x))=\varphi(st,x)\) holds for \(s,t\in S\) and \(x\in X\). With every function \(\alpha: S\rightarrow R\) the authors associate the \(\varphi\)-convolution \(\odot_\varphi\) by putting \[ (\alpha\odot_\varphi f)(x)=\sum_{s\in S}\alpha(s)f(\varphi(s,x)) \] for such functions \(f: X\rightarrow M\) for which the series on the right hand-side converges. The authors study this convolution and prove an abstract form of Möbius inversion formula. They give several examples of the usefulness of that formula, obtaining i.a. a simple proof of the Iseki-Tatuzawa formula, used in the elementary proof of the Prime Number Theorem.