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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.
Definition μ(n) is defined for all strictly positive natural numbers n and has its values in 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 50 first values of the function are plotted below 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: 0&mbox n>1end ight. (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, defined by 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. If n is a sphenic number (i.e. a product of three distinct primes), then clearly μ(n) = −1. The Lambert series for the Möbius function is μ(n) sections μ(n) = 0 if and only if n is divisible by a square. The first numbers with this property are (sequence in the On-Line Encyclopedia 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 (): 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 (): 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. Physics The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry. In this theory, the fundamental particles or "primons" have energies . Under second-quantization, multiparticle excitations are considered; these are given by 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. If they are taken as fermions, then the Pauli exclusion principle excludes squares. The operator (−1)''F'' which distinguishes fermions and bosons is then none other than the Möbius function . The free Riemann gas has a number of other interesting connection to number theory, including the fact that the partition function is the Riemann zeta function. This was used by Alain Connes in an attempted proof of the Riemann hypothesis. | ||||||||
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