Some Möbius-type functions and inversions constructed via difference operators (Q2724883)

Let a Möbius inversion pair \((f,g)\) of arithmetic functions \(f\) and \(g\) be defined by the Möbius inversion formulas NEWLINE\[NEWLINEf(n)= \sum_{d|n} g(d) \iff g(n)= \sum_{d|n} f(d) \mu \biggl(\frac{n}{d} \biggr)NEWLINE\]NEWLINE respectively (using the notation of Dirichlet convolution) \(f=g*1 \Leftrightarrow g=f*\mu\).NEWLINENEWLINENEWLINEThe authors show that such pairs can be expressed as simple reciprocal pairs of difference equations. Using this fact, they prove that the Möbius inversion formulas represent a discrete analogue of the Newton-Leibniz fundamental formulas. Applying some concepts of nonstandard analysis, the authors present some extensions of Möbius-type functions and they obtain more general Möbius-type inversions.