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| {{Otheruses4|the number-theoretic Möbius function|the combinatorial Möbius function|incidence algebra}}
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| :''For the [[rational function]]s defined on the [[complex number]]s, see [[Möbius transformation]].''
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| The classical '''Möbius function''' [[Mu_(letter)|''μ'']](''n'') is an important [[multiplicative function]] in [[number theory]] and [[combinatorics]]. The German mathematician [[August Ferdinand Möbius]] introduced it in 1832.<ref>Hardy & Wright, Notes on ch. XVI: "... μ(''n'') occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically."</ref><ref>In the ''[[Disquisitiones Arithmeticae]]'' (1801) [[Carl Friedrich Gauss]] showed that the sum of the primitive roots (mod ''p'') is μ(''p'' − 1), (see [[#Properties and applications]]) but he didn't make further use of the function. In particular, he didn't use Möbius inversion in the ''Disquisitiones''.</ref> This classical Möbius function is a special case of a more general object in combinatorics ([[#Generalizations|see below]]).
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| ==Definition==
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| ''μ''(''n'') is defined for all positive [[integer]]s ''n'' and has its values in <nowiki>{</nowiki>{{num/neg|1}}, {{num|0}}, {{num|1}}} depending on the [[integer factorization|factorization]] of ''n'' into [[prime factor]]s. It is defined as follows:
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| * ''μ''(''n'') = 1 if ''n'' is a [[square-free integer|square-free]] positive integer with an [[even and odd numbers|even]] number of prime factors.
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| * ''μ''(''n'') = −1 if ''n'' is a square-free positive integer with an odd number of prime factors.
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| * ''μ''(''n'') = 0 if ''n'' has a squared prime factor.
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| Values of ''μ''(''n'') for the first 25 positive numbers {{OEIS|id=A008683}}:
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| :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, ...
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| The first 50 values of the function are plotted below:
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| [[File:Moebius mu.svg|center|The 50 first values of the function]]
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| ==Properties and applications==
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| === Properties ===
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| The Möbius function is [[multiplicative function|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:
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| :<math>\sum_{d | n} \mu(d) = \begin{cases}1&\mbox{ if } n=1\\
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| 0&\mbox{ if } n>1.\end{cases}</math>
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| (A consequence of the fact that every non-empty finite set has just as many subsets with odd numbers of elements as subsets with even numbers of elements – in the same way as binomial coefficients exhibit alternating entries of odd and even power which sum symmetrically.) 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.
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| Other applications of ''μ''(''n'') in combinatorics are connected with the use of the [[Pólya enumeration theorem]] in combinatorial groups and combinatorial enumerations.
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| In number theory another [[arithmetic function]] closely related to the Möbius function is the [[Mertens function]], defined by
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| :<math>M(n) = \sum_{k = 1}^n \mu(k)</math>
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| 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]].
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| There is a formula<ref>{{harvnb|Hardy|Wright|1980|loc=(16.6.4), p. 239}}</ref> for calculating the Möbius function without directly knowing the factorization of its argument:
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| :<math>\mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,\,n)=1}} e^{2\pi i \tfrac{k}{n}},</math>
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| i.e. μ(''n'') is the sum of the primitive ''n''<sup>th</sup> [[roots of unity]]. (However, the computational complexity of this definition is at least the same as of the Euler Product definition.)
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| From this it follows that the Mertens function is given by:
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| :<math>M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a}</math> where <math> \mathcal{F}_n</math> is the [[Farey sequence]] of order ''n''.
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| This formula is used in the proof of the [[Farey_sequence#Riemann_hypothesis|Franel–Landau theorem]].<ref>Edwards, Ch. 12.2</ref>
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| The infinite symmetric matrix starting:
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| :<math> T = \begin{bmatrix} 1&1&1&1&1&1 \\ 1&-1&1&-1&1&-1 \\ 1&1&-2&1&1&-2 \\ 1&-1&1&-1&1&-1 \\ 1&1&1&1&-4&1 \\ 1&-1&-2&-1&1&2 \end{bmatrix} </math>
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| defined by the recurrence:
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| :<math> T(n,1)=1,\;T(1,k)=1,\;n \geq k:T(n,k) = -\sum\limits_{i=1}^{k-1} T(n-i,k),\;n<k: T(n,k)= -\sum\limits_{i=1}^{n-1} T(k-i,n) </math>
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| or:
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| :<math> T(n,k)=a(\gcd(n,k)) </math>
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| where "a" is the Dirichlet inverse of the Euler totient function,
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| can be used to calculate the Möbius function:<ref>Mats Granvik, ''[http://math.stackexchange.com/questions/84177/is-this-sum-equal-to-the-mobius-function Is this sum equal to the Möbius function?]'' (2011)</ref>
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| :<math> \mu(n) = \frac{1}{n} \sum\limits_{k=1}^{k=n} T(n,k) \cdot e^{2 \pi i \frac{k}{n}}. </math>
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| === Applications ===
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| ==== Mathematical series ====
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| The [[Dirichlet series]] that [[Generating function|generates]] the Möbius function is the (multiplicative) inverse of the [[Riemann zeta function]]
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| :<math>\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.</math>
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| This may be seen from its [[Euler product]]
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| :<math>\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)\cdots.</math>
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| The Dirichlet series for the Mobius function has the equivalence:
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| :<math>\sum_{n=1}^\infty \frac{\mu(n)}{n^{s}} = 1 - \sum_{a=2}^\infty \frac{1}{a^{s}} + \sum_{a=2}^\infty \sum_{b=2}^\infty \frac{1}{(a \cdot b)^{s}} - \sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \frac{1}{(a \cdot b \cdot c)^{s}} + \sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \sum_{d=2}^\infty \frac{1}{(a \cdot b \cdot c \cdot d)^{s}} - \cdots.</math>
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| The [[Lambert series]] for the Möbius function is:
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| :<math>\sum_{n=1}^\infty \frac{\mu(n)q^n}{1-q^n} = q.</math>
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| The ordinary generating function for the Möbius function follows from the binomial series
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| :<math>(I+X)^{-1}</math>
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| applied to triangular matrices:
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| :<math>\sum_{n=1}^\infty \mu(n)x^n = x - \sum_{a=2}^\infty x^{a} + \sum_{a=2}^\infty \sum_{b=2}^\infty x^{ab} - \sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty x^{abc} + \sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \sum_{d=2}^\infty x^{abcd} - \cdots</math>
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| ==== Algebraic number theory ====
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| Gauss<ref>Gauss, ''Disquisitiones'', Art. 81</ref> proved that for a prime number ''p'' the sum of its [[Primitive_root_modulo_n#Arithmetic_facts|primitive roots]] is congruent to μ(''p'' − 1) (mod ''p'').
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| If '''F'''<sub>''q''</sub> denotes the [[finite field]] of order ''q'' (where ''q'' is necessarily a prime power), then the number ''N'' of monic irreducible polynomials of degree ''n'' over '''F'''<sub>''q''</sub> is given by:<ref>{{harvnb|Jacobson|2009|loc=§4.13}}</ref>
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| :<math>N(q,n)=\frac{1}{n}\sum_{d|n} \mu(d)q^{\frac{n}{d}}.</math>
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| ==Average order==
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| The [[average order of an arithmetic function|average order]] of the Möbius function is zero. This statement is, in fact, equivalent to the [[prime number theorem]].<ref>{{harvnb|Apostol|1976|loc=§3.9}}</ref>
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| ==''μ''(''n'') sections==
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| ''μ''(''n'') = 0 [[if and only if]] ''n'' is divisible by the square of a prime. The first numbers with this property are {{OEIS|id=A013929}}:
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| :4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63,....
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| 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 three distinct prime factors ([[sphenic number]]s) are:
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| :30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, … {{OEIS|id=A007304}}.
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| and the first such numbers with 5 distinct prime factors are:
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| :2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, … {{OEIS|id=A046387}}.
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| ==Generalizations==
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| ===Incidence algebras===
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| 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 [[divisor|divisibility]]. See the article on [[incidence algebra]]s for the precise definition and several examples of these general Möbius functions.
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| ===Popovici's function===
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| Popovici defined a generalised Möbius function <math>\mu_k = \mu \star \cdots \star \mu</math> to be the ''k''-fold [[Dirichlet convolution]] of the Möbius function with itself. It is thus again a multiplicative function with
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| :<math> \mu_k(p^a) = (-1)^a \binom{k}{a} \ </math>
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| where the binomial coefficient is taken to be zero if ''a'' > ''k''. The definition may be extended to complex ''k'' by reading the binomial as a polynomial in ''k''.<ref name=HBNTII107>Sandor & Crstici (2004) p.107</ref>
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| ==Physics==
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| 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 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.
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| In the free Riemann gas, any natural number can occur, if the [[primon gas|primon]]s are taken as [[boson]]s. If they are taken as [[fermion]]s, then the [[Pauli exclusion principle]] excludes squares. The operator [[(-1)^F|(−1)<sup>''F''</sup>]] that distinguishes fermions and bosons is then none other than the Möbius function ''μ''(''n'').
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| The free Riemann gas has a number of other interesting connections to number theory, including the fact that the [[partition function (statistical mechanics)|partition function]] is the [[Riemann zeta function]]. This idea underlies [[Alain Connes]]' attempted proof of the [[Riemann hypothesis]].<ref> J.-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.</ref>
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| ==See also==
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| *[[Mertens function]]
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| *[[Liouville function]]
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| *[[Ramanujan's sum]]
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| *[[Sphenic number]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
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| *{{citation |last1=Gauss |first1=Carl Friedrich |authorlink1=Carl Friedrich Gauss |others=Arthur A. Clarke (English translator) | title=Disquisitiones Arithemeticae |edition=corrected 2nd |publisher=[[Springer Science+Business Media|Springer]] |location=New York |year=1986 |isbn=0-387-96254-9}}
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| *{{citation |last1=Gauss |first1=Carl Friedrich |authorlink1=Carl Friedrich Gauss |others=H. Maser (German translator) |title=Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) |edition=2nd |publisher=Chelsea |location=New York |year=1965 |isbn=0-8284-0191-8}}
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| *{{Apostol IANT}}
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| *[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.em/1047565447&view=body&content-type=pdf_1 Computing the summation of the Möbius function by Marc Deléglise and Joël Rivat] Experimental Mathematics Volume 5, Issue 4291-295
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| *{{Citation |last=Edwards |first=Harold |authorlink=Harold Edwards (mathematician) |title=Riemann's Zeta Function |publisher=Dover |location=Mineola, New York |date=1974 |isbn=0-486-41740-9}}
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| *{{citation |last1=Hardy |first1=G. H. |last2=Wright |first2=E. M. |title=An Introduction to the Theory of Numbers |edition=5th |publisher=[[Oxford University Press]] |location=Oxford |date=1980 |isbn=978-0-19-853171-5}}
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| *{{citation |last=Jacobson |first=Nathan |authorlink=Nathan Jacobson |title=Basic algebra I |year=2009 |edition=2nd |publisher=Dover Publications |isbn=978-0-486-47189-1 |origyear=1985}}
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| * {{citation | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=187–226 }}
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| * {{citation | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001 }}
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| *{{springer|author=N.I. Klimov|id=m/m064280|title=Möbius function}}
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| *[[Ed Pegg, Jr.]], "[http://www.maa.org/editorial/mathgames/mathgames_11_03_03.html The Möbius function (and squarefree numbers)]", ''MAA Online Math Games'' (2003)
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| {{refend}}
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| ==External links==
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| *{{mathworld|urlname= MoebiusFunction|title=Möbius function}}
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| *http://ghmath.wordpress.com/2010/06/20/recursive-relation-for-the-mobius-function/
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| *http://terrytao.wordpress.com/2008/07/13/the-mobius-and-nilsequences-conjecture/
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| {{DEFAULTSORT:Mobius Function}}
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| [[Category:Multiplicative functions]]
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