# 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

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.