Möbius Inversion Formula

The transform inverting the sequence

(1)

into

(2)

where the sums are over all possible integers  that divide  and  is the Möbius function.

The logarithm of the cyclotomic polynomial

(3)

is closely related to the Möbius inversion formula.

REFERENCES:

Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 91-93, 1979.

Jones, G. A. and Jones, J. M. "The Möbius Inversion Formula." §8.3 in Elementary Number Theory. Berlin: Springer-Verlag, pp. 148-152, 1998.

Hunter, J. Number Theory. London: Oliver and Boyd, 1964.

Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, pp. 577-580, 1974.

Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 28-29, 1951.

Schroeder, M. R. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. 19-20, 2000.

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 7-8 and 223-225, 1991.