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Wikipedia / Möbius function

The is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota.

Definition

The Möbius function is defined by

The Möbius function can alternatively be represented as

is the Kronecker delta is the Liouville function are the Prime omega functions, counted with multiplicity.

Another characterization by Carl Friedrich Gauss is the sum of all primitive roots.

Values

for the first 60 positive numbers areThe first 50 values of the function are plotted below:

Larger values can be checked in:

Applications

Mathematical series

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

This may be seen from its Euler product

Also:

The Lambert series for the Möbius function is

, we also have

Algebraic number theory

Gauss the sum of its primitive roots.

denotes the finite field is necessarily a prime power is given by

The Möbius function is used in the Möbius inversion formula.

Physics

The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry. Under second quantization for any natural number. This follows from the fact that the factorization of the natural numbers into primes is unique.

In the free Riemann gas, any natural number can occur, if the primons are taken as bosons. If they are taken as fermions, then the Pauli exclusion principle excludes squares. The operator (-1)^F.

The free Riemann gas has a number of other interesting connections to number theory, including the fact that the partition function is the Riemann zeta function. This idea underlies Alain Connes's attempted proof of the Riemann hypothesis.

Properties

The Möbius function is multiplicative are coprime). Proof, so

:

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

in combinatorics are connected with the use of the Pólya enumeration theorem in combinatorial groups and combinatorial enumerations.

There is a formula for calculating the Möbius function without directly knowing the factorization of its argument:

-th roots of unity. (However, the computational complexity of this definition is at least the same as that of the Euler product definition.)

Other identities satisfied by the Möbius function include

and

  1. .

The first of these is a classical result while the second was published in 2020. Similar identities hold for the Mertens function.

\mu over divisors

The formula

can be written using Dirichlet convolution is the identity under the convolution.

One way of proving this formula is by noting that the Dirichlet convolution of two multiplicative functions

  1. ,

    .

Other proofs

Another way of proving this formula is by using the identity

.

. Then there is a bijection. The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets.

. The inductive step follows directly from these two bijections.

A related result is that the binomial coefficients exhibit alternating entries of odd and even power which sum symmetrically.

Average order

The mean value (in the sense of average orders) of the Möbius function is zero. This statement is, in fact, equivalent to the prime number theorem.

\mu(n) sections

if and only if is divisible by the square of a prime. The first numbers with this property are

  1. 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, ... .

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

  1. 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

  1. 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, ... .

Mertens function

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

for every natural number . This function is closely linked with the positions of zeroes of the Riemann zeta function. See the article on the Mertens conjecture and the Riemann hypothesis.

From the formula

it follows that the Mertens function is given by

is the Farey sequence.

This formula is used in the proof of the Franel–Landau theorem.

Generalizations

Incidence algebras

In combinatorics, every locally finite partially ordered set (poset) is assigned an incidence algebra. One distinguished member of this algebra is that poset's "Möbius function". The classical Möbius function treated in this article is essentially equal to the Möbius function of the set of all positive integers partially ordered by divisibility. See the article on incidence algebras for the precise definition and several examples of these general Möbius functions.

Because the Möbius function is multipliciative, so is its (iterated) Dirichlet convolution-fold Dirichlet convolution where the binomial coefficient..

Implementations

See also

Notes

Citations

Sources