Möbius series

A series of functions of the form

(*)

These series were investigated by A. Möbius [1], who found for a series (*) the inversion formula

where is the Möbius function. Möbius considered also inversion formulas for finite sums running over the divisors of a natural number :

Another inversion formula: If is a totally-multiplicative function (cf. Multiplicative arithmetic function) for which , and is a function defined for all real , then

implies

References

[1]	A. Möbius, "Ueber eine besondere Art der Umkehrung der Reihen" J. Reine Angew. Math. , 9 (1832) pp. 105–123
[2]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[3]	K. Prachar, "Primzahlverteilung" , Springer (1957)

Comments

All these (and many other) inversion formulas follow from the basic property of the Möbius function that it is the inverse of the unit arithmetic function under the convolution product, cf. (the editorial comments to) Möbius function and Multiplicative arithmetic function.