# Möbius function

**Moebius functionMobius functionμ(''n'')Mu functionMöbius mu functionMöbius sumMöbius μ functionμ(12)**

The classical '''Möbius functionwikipedia

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### Multiplicative function

**multiplicativecompletely multiplicativeMultiplicativity**

''' is an important multiplicative function in number theory and combinatorics.

### Mertens function

**M''(12)**

In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by

where μ(k) is the Möbius function.

### Möbius inversion formula

**Möbius inversionMöbius transformMoebius inversion**

The equality above leads to the important Möbius inversion formula and is the main reason why μ is of relevance in the theory of multiplicative and arithmetic functions.

where μ is the Möbius function and the sums extend over all positive divisors d of n (indicated by d \mid n in the above formulae).

### Liouville function

**Liouville's functionλ''(''n'')**

is the Liouville function,

If n is squarefree, i.e., if where p_i is prime for all i and where, then we have the following alternate formula for the function expressed in terms of the Möbius function and the distinct prime factor counting function \omega(n):

### Arithmetic function

**arithmetic functionsarithmetical functionnumber-theoretic function**

In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by

μ(n), the Möbius function, is important because of the Möbius inversion formula.

### August Ferdinand Möbius

**MöbiusAugust MöbiusAugust F. Möbius**

The German mathematician August Ferdinand Möbius introduced it in 1832.

His interest in number theory led to the important Möbius function μ(n) and the Möbius inversion formula.

### Riemann hypothesis

**Critical line theoremcritical line1st**

and the Riemann hypothesis.

where μ is the Möbius function.

### Square-free integer

**square-freesquarefreesquare-free number**

is a square-free positive integer with an even number of prime factors.

A positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.

### Dirichlet series

**Dirichlet-SeriesFormal Dirichlet serieszeta**

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have

where μ(n) is the Möbius function.

### Mertens conjecture

See the article on the Mertens conjecture for more information about the connection between

where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1,

### Root of unity

**roots of unityprimitive root of unitycyclotomy**

as the sum of the primitive nth roots of unity.

is the Möbius function.

### Lambert series

**previous section**

The Lambert series for the Möbius function is:

Möbius function given below \mu(n)

### Riemann zeta function

**zeta functionRiemann zeta-functionRiemann's zeta function**

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function.

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function

### Farey sequence

**Farey graphFarey DiagramFarey series**

where n is the Farey sequence of order n.

:where µ(d) is the number-theoretic Möbius function, and is the floor function.

### Euler product

**Euler factorEuler's product formulaproduct of Euler factors**

This may be seen from its Euler product

Using their reciprocals, two Euler products for the Möbius function \mu(n) are

### Average order of an arithmetic function

**average orderaverage order summatory functions**

The average order of the Möbius function is zero.

, the Möbius function, is zero; this is again equivalent to the prime number theorem.

### Primitive root modulo n

**primitive rootprimitive rootsPrimitive root modulo ''n**

Gauss proved that for a prime number p the sum of its primitive roots is congruent to

modulo p, where μ is the Möbius function.

### Finite field

**Galois fieldfinite fieldsGF**

denotes the finite field of order q (where q is necessarily a prime power), then the number N of monic irreducible polynomials of degree n over

is the Möbius function.

### Incidence algebra

**Möbius functiondenote thisgeneralized Möbius function**

In combinatorics, every locally finite partially ordered set (poset) is assigned an incidence algebra.

The incidence algebra zeta function ζ D (a,b) = 1 corresponds to the classical Riemann zeta function having reciprocal where is the classical Möbius function of number theory.

### Ramanujan's sum

**Ramanujan sumhere**

\mu(n) is the Möbius function, and

### Dirichlet convolution

**Dirichlet inverseDirichlet ringconvolution**

to be the k-fold Dirichlet convolution of the Möbius function with itself.

### Prime number theorem

**distribution of primesdistribution of prime numbersprime number theorem for arithmetic progressions**

This statement is, in fact, equivalent to the prime number theorem.

is the Möbius function.

### Sphenic number

**sphenic numbers**

The first such numbers with three distinct prime factors (sphenic numbers) are:

The Möbius function of any sphenic number is −1.

### Primon gas

**Free Riemann gas**

The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry.

The fermion operator (−1) F has a very concrete realization in this model as the Möbius function \mu(n), in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.

### Number theory

**number theoristcombinatorial number theorytheory of numbers**

''' is an important multiplicative function in number theory and combinatorics.