NumberTheory
Moebius
Moebius function
Calling Sequence
Parameters
Description
Examples
Compatibility
Moebius(n)
Möbius(n)
mu(n)
μn
n
-
positive integer
The Moebius(n) command computes the Moebius function of the positive integer n.
If n is divisible by the square of a prime number, then Moebius(n) is equal to 0. Otherwise, Moebius(n) is equal to 1 if n has an even number of prime factors, and is equal to -1 if n has an odd number of prime factors.
Möbius and mu are aliases of Moebius.
You can enter the command mu using either the 1-D or 2-D calling sequence. For example, mu(8) is equivalent to μ8.
withNumberTheory:
Moebius1
1
Moebius33⋅5
0
Moebius3⋅5⋅7
−1
Moebius23⋅11
The Möbius function is multiplicative as an arithmetic function. That is, if n and m are coprime then Moebius(n*m) = Moebius(n)*Moebius(m).
igcd5657,31945103
μ5657⋅31945103
μ5657μ31945103
The first 50 values for the Moebius function are plotted below:
plots:-pointplotseqn,Moebiusn,n=1..50,labels=n,μn,symbol=soliddiamond,symbolsize=15,color=OrangeRed,size=600,400,tickmarks=default,−1,0,1
The NumberTheory[Moebius] command was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
See Also
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