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
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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 Formula
Cite this as:
Weisstein, Eric W. "Möbius Inversion Formula."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MoebiusInversionFormula.html
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