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The classical Möbius function is an important multiplicative function in number theory and combinatorics. It is named in honor of the German mathematician August Ferdinand Möbius, who first introduced it in 1831. This classical Möbius function is a special case of a more general object in combinatorics. In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...
Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...
Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ...
August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ...
Definition
μ(n) is defined for all positive natural numbers n and has its values in {−1, 0, 1} depending on the factorization of n into prime factors. It is defined as follows Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In mathematics, −1 is the integer greater than negative two (−2) and less than 0. ...
Zero redirects here. ...
One redirects here. ...
In mathematics, the integer prime-factorization (also known as prime decomposition) problem is this: given a positive integer, write it as a product of prime numbers. ...
In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...
- μ(n) = 1 if n is a square-free positive integer with an even number of distinct prime factors.
- 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, but the Maple computer algebra system for example returns −1 for this value. In mathematics, a square-free integer is one divisible by no perfect square, except 1. ...
In mathematics, any integer (whole number) is either even or odd. ...
In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ...
In mathematics, any integer (whole number) is either even or odd. ...
Maple is a general purpose commercial computer algebra system. ...
Properties and applications The Möbius function is multiplicative (i.e. μ(ab) = μ(a)μ(b) whenever a and b are coprime). The sum over all positive divisors of n of the Möbius function is zero except when n = 1: In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...
In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
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(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 that μ is of relevance in the theory of multiplicative and arithmetic functions.
Other applications of μ(n) in combinatorics are connected with the use of the Polya theorem in combinatorial groups and combinatorial enumerations.
In number theory another arithmetic function closely related to the Möbius function is the Mertens function; it is defined by: In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ...
In number theory, the Mertens function is where μ(k) is the Möbius function. ...
for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function. See the article on the Mertens conjecture for more information about the connection between M(n) and the Riemann hypothesis. In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...
This article needs cleanup. ...
RH directs here. ...
If n is a sphenic number (i.e. a product of three distinct primes), then clearly μ(n) = −1. A sphenic number is a positive integer that is the product of three distinct prime factors. ...
μ(n) sections μ(n) = 0 if and only if n is divisible by a square. The first numbers with this property are (sequence A013929 in the On-Line Encyclopedia of Integer Sequences): In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...
The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...
4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63,... 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 (OEIS:A007304): A sphenic number is a positive integer that is the product of three distinct prime factors. ...
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 (OEIS:A046387): 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, ...
Generalization In combinatorics, every locally finite 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. Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ...
In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ...
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