Generalizations and Extensions of the Möbius Function

Todorka Nedeva

Monday, February 23, 2004
1:00 pm, 845 Patterson Office Tower

Abstract:

The classical Möbius function appeared in Euler's work (1748) with the consideration of the Riemann zeta function. However, its arithmetical importance was first recognized by A. F. Möbius in 1832 with the discovery of a number of inversion formulae. The first rigorous treatment of the inversion problem of Möbius was given by E. Hille and O. Szász (1936/37), while the finite form of the Möbius inversion formula was discovered simultaneously by R. Dedekind and J. Liouville in 1857.

In 1964, G.-C. Rota introduced a general theory of Möbius function on partially ordered sets. This theory includes the principle of inclusion-exclusion and Möbius inversion as special cases. The Möbius function has many generalizations in Number Theory. Analogous functions arise from certain divisibility or product notions of arithmetic functions. More interestingly Möbius functions and inversion theorems appear ouside the number theoretical setting, e.g. Group Theory, Lattice Theory, Partially Ordered Sets and Arithmetical Semigroups.