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NumberTheory

  

Moebius

  

Moebius function

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Moebius(n)

 

Möbius(n)

 

mu(n)

μn

Parameters

n

-

positive integer

Description

• 

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.

Examples

withNumberTheory:

Moebius1

1

(1)

Moebius335

0

(2)

Moebius357

−1

(3)

Moebius2311

1

(4)

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

1

(5)

μ565731945103

−1

(6)

μ5657μ31945103

−1

(7)

The first 50 values for the Moebius function are plotted below:

plots:-pointplotseqn,Moebiusn,n=1..50,labels=n,`μ`n,symbol=soliddiamond,symbolsize=15,color=OrangeRed,size=600,400,tickmarks=default,1,0,1

Compatibility

• 

The NumberTheory[Moebius] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

NumberTheory