Mobius function

The Möbius function μ(n), named after August Ferdinand Möbius, is a multiplicative function studied in number theory and combinatorics.

Definition

μ(n) is defined for all positive integers n and has values in {-1, 0, 1} depending on the factorization of n on its prime factors. It is defined as follows:

• μ(n) = 1 yes n is free of squares and has a number of prime factors.
• μ(n= -1 yes n is free of squares and has an odd number of prime factors.
• μ(n) = 0 yes n It's divisible by some square.

An equivalent definition is defined using the functions ω(n) and Ω(n), where:

• ω(n) obtains the number of different cousins that divide the number.
• Ω(n) gets the number of prime factors nincluding its multiply. Clearly, ω(n) ≤ Ω(n).

Thus, the Möbius function is defined as

<math alttext="{displaystyle mu (n)={begin{cases}(-1)^{omega (n)}=(-1)^{Omega (n)}&{mbox{si }};omega (n)=Omega (n)\0&{mbox{si }};omega (n)μ μ (n)={(− − 1)ω ω (n)=(− − 1)Ω Ω (n)Yeah.ω ω (n)=Ω Ω (n)0Yeah.ω ω (n).Ω Ω (n).{displaystyle mu (n)={begin{cases}(-1)^{omega (n)}=(-1)^{Omega (n)}{mbox{si }}}{omega (n)=omega (n) supposed{mbox{si }}{omega (n)}{omega (n).<img alt="{displaystyle mu (n)={begin{cases}(-1)^{omega (n)}=(-1)^{Omega (n)}&{mbox{si }};omega (n)=Omega (n)\0&{mbox{si }};omega (n)

The definition implies that μ(1) = 1, since 1 has 0 distinct prime factors, hence an even number.

Representation

The table of values of μ(n) for the first twenty positive integers (sequence A008683 in OEIS) is:

The first 50 values of the function μ(n), represented in the following graph:

Properties and applications

The Möbius function is multiplicative, and has great relevance in the theory of multiplicative and arithmetic functions since it appears in the Möbius inversion formula. The sum over all positive divisors of n of the Möbius function is zero except when n = 1.

1.end{cases}}}" xmlns="http://www.w3.org/1998/Math/MathML">␡ ␡ d日本語nμ μ (d)={1Yeah.n=10Yeah.n▪1.{displaystyle sum _{dsharing}mu (d)={begin{cases}1 fake{mbox{ if }n=1 stranger{mbox{mbox{ if }}n regula1.end{cases}}}}}}1.end{cases}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa9a398dd6ae0a89c9d901de19f2cc2d005ae3e" style="vertical-align: -3.505ex; width:26.274ex; height:7.176ex;"/>

Other applications of μ(n) in combinatorics are related to the use of Pólya's theorem on combinatorial groups.

Number Theory

In number theory, the Mertens function is related to the Möbius function, and is defined as:

M(n)=␡ ␡ 1≤ ≤ k≤ ≤ nμ μ (k){displaystyle M(n)=sum _{1leq kleq n}mu (k)}}

for every natural number n. This function is related to the positions of the zeros of the Euler-Riemann function ζ and to the Riemann conjecture.