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.
See also
Cyclotomic Polynomial,
Dirichlet Generating Function,
Möbius Function,
Möbius Transform Explore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Möbius Inversion Formula Cite this as:
Weisstein, Eric W. "Möbius Inversion Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MoebiusInversionFormula.html
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