The Mobius Function μ (n) : Number Theory #The #Mobius #Function #μ (n) : #Number #Theory

February 27, 2026

Number theory , like many other branches of mathematics , is often concerned with sequences of real or complex numbers. In number theory such sequences are called arithmetic functions.

Definition :

A real or complex - valued functions defined on the positive integers is called an arithmetical function or a number- theoretic function.

The Mobius Function μ (n) :

The Mobius function μ is defined as follows :

μ(1) = 1 ;

If n > 1 , write n = p₁^a1 + p₂ ^a2 + … + p_k ^ak. Then

μ( n) = ( -1 )^k if a₁ = a₂= … = a_k =1

= 0 otherwise

Note :

μ(n) = 0 if and only if n has a square factor > 1.

For example :

We have μ(1) = 1.

For n = 2, we have n = 2 = 2¹ , μ( 2 ) = ( -1 )^k = ( -1 )¹ = -1.

For n = 3 , we have n = 3 = 3¹ , μ( 3 ) = ( -1 )^k = ( -1 )¹ = -1.

For n = 4, we have n = 4 = 2² = 2^a , μ( 4 ) = 0. | since a ≠ 1 |

For n = 5 , we have n = 5 = 5¹ , μ( 5 ) = ( -1 )^k = ( -1 )¹ = -1.

For n = 6, we have n = 6 = 2¹ x 3¹= 2^a1 x 3^a2= ( -1 ) ² = 1 .

For n = 7 , we have n = 7 = 7¹ , μ( 7 ) = ( -1 )^k = ( -1 )¹ = -1.

For n = 8, we have n = 8 = 2³ = 2^a , μ( 8 ) = 0 | since a ≠ 1 |

Here is a short table of values of μ(n).

n :	1	2	3	4	5	6	7	8	9	10
μ (n) :	1	-1	-1	0	-1	1	-1	0	0	1

The Mobius function arises in many different places in number theory. One of its fundamental properties is a remarkably simple formula for the divisor sum Σ μ(d) ( for d/n ) extended over the positive divisors of n. In this formula , [ x ] denotes the greatest integer ≤ x.

Note :

If n ≥ 1 we have

Σ μ(d) = [1/n ] = 1 if n = 1

= 0 if n > 1

* Euclidean Algorithm

* Fundamental Theorem of Arithmetic

* Properties of Numbers

* Historical Introduction to Number Theory

* GCD of morethan 2 numbers