Generalizations and extensions of the möbius function

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2.1 Introduction

In this chapter we will survey many generalizations in Number theory of the Möbius function, as well as analogues functions which arose by the extensions of certain divisibility notions or product notions of arithmetical functions. Our study will include also Möbius functions in Group theory, Lattice theory, Partially ordered sets or Arithmetical semigroups. Sometimes references to applications will be pointed out, too. This is a refined and extended version of [164].

The classical Möbius function is defined by (1)

The function occurred implicitly in L. Euler ([67] paragraph 269) with the considerations of the “Riemann zeta function”

and the reciprocal formula (2)

However, its arithmetical importance was first recognized by A. F. Möbius [143] in 1832, with the discovery of a number of inversion formulae.

Möbius raised the following question:

Given an arbitrary function f(z) and g(z) by ...