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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

with(NumberTheory):

Moebius(1);

1

(1)

Moebius(3^3 * 5);

0

(2)

Moebius(3*5*7);

−1

(3)

Moebius(23*11);

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).

igcd(5657, 31945103);

1

(5)

mu(5657 * 31945103);

−1

(6)

mu(5657) * mu(31945103);

−1

(7)

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

plots:-pointplot([seq([n, Moebius(n)], 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