Möbius function

Definition

The Möbius function μ is an important multiplicative function considered in number theory and in combinatorics. μ(n) is defined for all positive natural numbers n and has its values in {-1, 0, 1} depending on the natural factorization of n. It is defined as follows

μ(n) = 1 if n is a square-free positive integer with an even number of distinct prime factors.
μ(n) = -1 if n is a square-free positive integer with an odd number of distinct prime factors.
μ(n) = 0 if n is not square-free.

This is taken to imply that μ(1) = 1. For n= 0 μ(n) by default is not defined. (For example Maple returns -1 for this value.)

The Möbius function is of relevance in the theory of multiplicative and arithmetic functions because it appears in the Möbius inversion formula.

Maple Calling Sequence notation:

> with(numtheory):
> mobius(n);

μ(n) sections

If μ(n) = 0, then n is said to be divisible by a square. The first numbers with this property are (Sloane ID Number A013929 ??):

4,8,9,12,16,18,20,24,25,27,28,32,36,40,44,45,48,49,50,52,54,56,60,63,...

If n is prime, then μ(n) = -1, but the converse is not true. The first non prime n for which μ(n) = -1 is 30 = 2*3*5. The first such numbers with 3 distinct prime factors are (SIDN A007304):

30,42,66,70,78,102,105,110,114,130,138,154,165,170,174,182,186,190,195,222,...

and the first such numbers with 5 distinct prime factors are (SIDN A046387):

2310,2730,3570,3990,4290,4830,5610,6006,6090,6270,6510,6630,7410,7590,7770,7854, 

8610,8778,8970,9030,9282,9570,9690,9870,10010,10230,10374,10626,11130,11310,11730, <br 
12090,12210,12390,12558,12810,13090,13110...

Very simple numbers to above ones are numbers by exactly 5 different primes, but they differ in values of μ(n) for it is μ(4620) = 0, as 4620 = 2 ². 3. 5. 7. 11 (SIDN A051270):

2310,2730,3570,3990,4290,4620,4830,5460,5610,6006,6090,6270,6510,6630,6930,7140, 

7410,7590,7770,7854,7980,8190,8580,8610,8778,8970,9030,9240,9282,9570,9660,9690, 

9870,10010,10230,10374,10626,10710,10920,11130,...

In number theory another arithmetic function closely related to the Möbius function is very important; it is defined by:

M(n) = ∑_1≤k≤n μ(k) .

for every natural number n. This function is closely linked with the positions of zeroes of Euler - Riemann ζ- function. The connection between M(n) and the Riemann conjecture was known to Thomas Joannes Stieltjes. See the article on the Mertens conjecture for more information about this connection.

μ(n) representations

If we treat μ(n) as a coloured L-system we get these kind of representations:

File:Mols001.gif Representation of μ(n) on a plane as a coloured L-system around fixed point n=1.

File:Mols002-100000.gif Representation of μ(n) on a plane as a coloured L-system around fixed point n=100,000.

If we examine μ(n) with connection of prime number spiral and Ulam's cloth (also called Ulam's spiral):

File:Ulmo000.gif Ulam's cloth on a plane with fixed point n=1.

we get representations as:

File:Ulmo001.gif Representation of μ(n) on a plane with fixed point n=1 together with Ulam's cloth.

File:Ulmo002.gif Representation of μ(n) on a plane with fixed point n=1 without Ulam's cloth.

Note:
Axel feel free to change the contents but keep in mind the original efforts on a subject.

See also:
-Similar representations of Ulam's cloth at: http://www.maths.ex.ac.uk/~mwatkins/zeta/ulam.htm