Möbius function - Citizendia

This article is about the number-theoretic Möbius function. For the combinatorial Möbius function, see incidence algebra. In Order theory, a field of Mathematics, an incidence algebra is an Associative algebra, defined for any locally finite Partially ordered set

The classical Möbius function $\!\,\mu(n)$ is an important multiplicative function in number theory and combinatorics. Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects It is named in honor of the German mathematician August Ferdinand Möbius, who first introduced it in 1831. August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and This classical Möbius function is a special case of a more general object in combinatorics (see below).

Definition

μ(n) is defined for all positive integers n and has its values in {−1, 0, 1} depending on the factorization of n into prime factors. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Number theory, the prime factors of a positive Integer are the Prime numbers that divide into that integer exactly without leaving a remainder It is defined as follows:

μ(n) = 1 if n is a square-free positive integer with an even number of distinct prime factors. In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 In Mathematics, the parity of an object states whether it is even or odd
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. The value of μ(0) is generally left undefined.

Values of μ(n) for the first 25 positive numbers (sequence A008683 in OEIS):

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, . The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences . .

The 50 first values of the function are plotted below:

Properties and applications

The Möbius function is multiplicative (i. Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative e. μ(ab) = μ(a) μ(b) whenever a and b are coprime). In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than The sum over all positive divisors of n of the Möbius function is zero except when n = 1:

$\sum_{d | n} \mu(d) = \left\{\begin{matrix}1&\mbox{ if } n=1\\0&\mbox{ if } n>1\end{matrix}\right.$

(A consequence of the fact that every non-empty finite set has just as many subsets with an even number of elements as it has subsets with an odd number of elements. ) This 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. In Mathematics, the classic Möbius inversion formula was introduced into Number theory during the 19th century by August Ferdinand Möbius.

Other applications of μ(n) in combinatorics are connected with the use of the Pólya enumeration theorem in combinatorial groups and combinatorial enumerations. "Enumeration theorem" redirects here For its labelled counterpart see Labelled enumeration theorem.

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

$M(n) = \sum_{k = 1}^n \mu(k)$

for every natural number n. In Number theory an arithmetic function or arithmetical function is a Function defined on the set of Natural numbers (i In Number theory, the Mertens function is M(n = \sum_{1\le k \le n} \mu(k where μ(k is the Möbius function. This function is closely linked with the positions of zeroes of the Riemann zeta function. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in See the article on the Mertens conjecture for more information about the connection between M(n) and the Riemann hypothesis. In Mathematics, the Mertens conjecture is a statement about the behaviour of a certain function as its argument increases The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved

The Lambert series for the Möbius function is

$\sum_{n=1}^\infty \frac{\mu(n)q^n}{1-q^n} = q.$

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function

$\sum_1^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.$

This is easy to see from its Euler product

$\frac{1}{\zeta(s)} = \prod_{p\in \mathbb{P}}{\left(1-\frac{1}{p^{s}}\right)}= \left(1-\frac{1}{2^{s}}\right)\left(1-\frac{1}{3^{s}}\right)\left(1-\frac{1}{5^{s}}\right)\dots.$

μ(n) sections

μ(n) = 0 if and only if n is divisible by a square. In Mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form S(q=\sum_{n=1}^\infty a_n \frac {q^n}{1-q^n} In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in In Number theory, an Euler product is an Infinite product expansion indexed by Prime numbers p, of a Dirichlet series. ↔ The first numbers with this property are (sequence A013929 in OEIS):

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 On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences . .

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 (sphenic numbers) are:

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, … (sequence A007304 in OEIS)

and the first such numbers with 5 distinct prime factors are:

2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, … (sequence A046387 in OEIS)

Generalization

In combinatorics, every locally finite partially ordered set (poset) is assigned an incidence algebra. In Mathematics, a sphenic number ( Old Greek sphen = Wedge) is a positive integer which is the product of three distinct Prime The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Order theory, a field of Mathematics, an incidence algebra is an Associative algebra, defined for any locally finite Partially ordered set 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. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without See the article on incidence algebras for the precise definition and several examples of these general Möbius functions. In Order theory, a field of Mathematics, an incidence algebra is an Associative algebra, defined for any locally finite Partially ordered set

Physics

The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry. In Mathematical physics, the primon gas or free Riemann gas is a Toy model illustrating in a simple way some correspondences between Number theory In Mathematical physics, the primon gas or free Riemann gas is a Toy model illustrating in a simple way some correspondences between Number theory In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In this theory, the fundamental particles or "primons" have energies $log p$ . Under second-quantization, multiparticle excitations are considered; these are given by $log n$ for any natural number n. 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. In Mathematical physics, the primon gas or free Riemann gas is a Toy model illustrating in a simple way some correspondences between Number theory In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein If they are taken as fermions, then the Pauli exclusion principle excludes squares. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 The operator (−1)^F which distinguishes fermions and bosons is then none other than the Möbius function $μ(n)$ .

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. In Statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in Thermodynamic In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in This idea underlies Alain Connes' attempted proof of the Riemann hypothesis. Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved ^[1]

References

^ J. In Mathematics, a sphenic number ( Old Greek sphen = Wedge) is a positive integer which is the product of three distinct Prime This article is not about Ramanujan summation. In Number theory, a branch of Mathematics, Ramanujan's sum, usually denoted -B. Bost and Alain Connes (1995), "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 411-457.

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0387901639
Ed Pegg, Jr., "The Möbius function (and squarefree numbers)", MAA Online Math Games (2003)
Eric W. Weisstein, Möbius function at MathWorld. Ed Pegg Jr is an expert on mathematical Puzzles and is a self-described recreational mathematician Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
N. I. Klimov (2001), “Möbius function”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Computing the summation of the Möbius function by Marc Deléglise and Joël Rivat Experimental Mathematics Volume 5, Issue 4291-295

The Encyclopaedia of Mathematics is a large reference work in Mathematics.