Möbius Inversion Formula
The transform inverting the sequence
| [画像: g(n)=sum_(d|n)f(d) ] |
(1)
|
into
where the sums are over all possible integers d that divide n and mu(d) is the Möbius function.
The logarithm of the cyclotomic polynomial
is closely related to the Möbius inversion formula.
See also
Cyclotomic Polynomial, Dirichlet Generating Function, Möbius Function, Möbius TransformExplore with Wolfram|Alpha
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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 FormulaCite this as:
Weisstein, Eric W. "Möbius Inversion Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MoebiusInversionFormula.html