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This classical Möbius function is a special case of a more general object in combinatorics.
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
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.
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:
(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 formulaNumber theory Combinatorics The classic Mobius inversion formula was introduced into number theory during the 19th century by August Ferdinand Mobius. It was later generalized to other "Mobius inversion formulas"; see incidence algebra. The classic versio 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 functionIn 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 other words: an arithmetic function is nothing but a sequence of complex numbers. closely related to the Möbius function is the Mertens functionIn number theory, the Mertens function is : where μ(k) is the Mobius function. Because the Mobius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no x such that M ''x > x''. The Me; it is defined by:
for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta functionIn 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. It also has applications in physics. Definition The Riemann zeta function ζ s is. See the article on the Mertens conjectureNumber theory Conjectures In number theory, if we define the Mertens function as : where μ(k) is the Mobius function, then the Mertens conjecture is that : Stieltjes claimed in 1885 to have proved that always stayed between two fixed bounds, but did no for more information about the connection between M(n) and the Riemann hypothesisThe Riemann hypothesis first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ s . It is one of the most important open problems of contemporary mathematics; a $1,000,000 prize ha.
If n is a sphenic numberA sphenic number is a positive integer that is the product of three distinct prime factors. The Mobius function returns -1 when passed any sphenic number. Note that this definition is more stringent than simply requiring the integer to have exactly three (i.e. a product of three distinct primes), then clearly μ(n) = −1.