Editing Möbius function

{{short description|Multiplicative function in number theory}}
{{About|the number-theoretic Möbius function|the combinatorial Möbius function|incidence algebra|the [[rational function]]s defined on the [[complex number]]s|Möbius transformation}}
{{use dmy dates|date=October 2024}}
{{More footnotes needed|date=October 2024}}
{{Infobox integer sequence
| image                 = 
| image_size            = 
| alt                   = 
| caption               = 
| named_after           = [[August Ferdinand Möbius]]
| publication_year      = 1832
| author                = [[August Ferdinand Möbius]]
| terms_number          = infinite
| con_number            =
| number                =
| parentsequence        =
| formula               =
| first_terms           = 1, −1, −1, 0, −1, 1, −1, 0, 0, 1
| largest_known_term    =
| OEIS                  = A008683
| OEIS_name             = Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.
}}
The '''Möbius function <math>\mu(n)</math>''' is a [[multiplicative function]] in [[number theory]] introduced by the German mathematician [[August Ferdinand Möbius]] (also transliterated ''Moebius'') in 1832.{{efn-lr|Hardy & Wright, Notes on ch. XVI: "... <math>\mu(n)</math> occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically". {{harv|Hardy|Wright|1980|loc=Notes on ch. XVI}}}}{{efn-lr|In the ''[[Disquisitiones Arithmeticae]]'' (1801) [[Carl Friedrich Gauss]] showed that the sum of the primitive roots (<math>\mod p</math>) is <math>\mu(p-1)</math>, (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''.{{sfn|Gauss|1986|loc=Art. 81}} 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reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.}}{{sfn|Möbius|1832|pp=105–123}} It is ubiquitous in elementary and [[analytic number theory]] and most often appears as part of its namesake the [[Möbius inversion formula]]. Following work of [[Gian-Carlo Rota]] in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted <math>\mu(x)</math>.

==Definition==
The Möbius function is defined by{{sfn|Abramowitz|Stegun|1972|p=826}}

:<math display="block">\mu(n) = 
\begin{cases}
1 & \text{if } n = 1 \\
(-1)^k & \text{if } n \text{ is the product of } k \text{ distinct primes} \\
0 & \text{if } n \text{ is divisible by a square} > 1
\end{cases}</math>

The Möbius function can alternatively be represented as

: <math>\mu(n) = \delta_{\omega(n)\Omega(n)} \lambda(n),</math>

where <math>\delta_{ij}</math> is the [[Kronecker delta]], <math>\lambda(n)</math> is the [[Liouville function]], 
[[Prime omega function|<math>\omega(n)</math>]] is the number of distinct prime divisors of <math>n</math>, and [[Prime omega function|<math>\Omega(n)</math>]] is the number of prime factors of <math>n</math>, counted with multiplicity.

Another characterization by [[Carl Friedrich Gauss]] is the sum of all [[Primitive root modulo n|primitive roots]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Möbius Function |url=https://mathworld.wolfram.com/MoebiusFunction.html |access-date=2024-10-01 |website=mathworld.wolfram.com |language=en}}</ref>

==Values==
The values of <math>\mu(n)</math> for the first 50 positive numbers are

{|class="wikitable" colwidth="2em" style="text-align:center"
|-
!<math>n</math>
! style="width: 30px;" |1
! style="width: 30px;" |2
! style="width: 30px;" |3
! style="width: 30px;" |4
! style="width: 30px;" |5
! style="width: 30px;" |6
! style="width: 30px;" |7
! style="width: 30px;" |8
! style="width: 30px;" |9
! style="width: 30px;" |10
|-
!<math>\mu(n)</math>
|1
|−1
|−1
|0
|−1
|1
|−1
|0
|0
|1
|}

{|class="wikitable" style="text-align:center"
|-
!<math>n</math>
! style="width: 30px;" |11
! style="width: 30px;" |12
! style="width: 30px;" |13
! style="width: 30px;" |14
! style="width: 30px;" |15
! style="width: 30px;" |16
! style="width: 30px;" |17
! style="width: 30px;" |18
! style="width: 30px;" |19
! style="width: 30px;" |20
|-
!<math>\mu(n)</math>
|−1
|0
|−1
|1
|1
|0
|−1
|0
|−1
|0
|}

{|class="wikitable" style="text-align:center"
|-
!<math>n</math>
! style="width: 30px;" |21
! style="width: 30px;" |22
! style="width: 30px;" |23
! style="width: 30px;" |24
! style="width: 30px;" |25
! style="width: 30px;" |26
! style="width: 30px;" |27
! style="width: 30px;" |28
! style="width: 30px;" |29
! style="width: 30px;" |30
|-
!<math>\mu(n)</math>
|1
|1
|−1
|0
|0
|1
|0
|0
|−1
|−1
|}

{|class="wikitable" style="text-align:center"
|-
!<math>n</math>
! style="width: 30px;" |31
! style="width: 30px;" |32
! style="width: 30px;" |33
! style="width: 30px;" |34
! style="width: 30px;" |35
! style="width: 30px;" |36
! style="width: 30px;" |37
! style="width: 30px;" |38
! style="width: 30px;" |39
! style="width: 30px;" |40
|-
!<math>\mu(n)</math>
|−1
|0
|1
|1
|1
|0
|−1
|1
|1
|0
|}

{|class="wikitable" style="text-align:center"
|-
!<math>n</math>
! style="width: 30px;" |41
! style="width: 30px;" |42
! style="width: 30px;" |43
! style="width: 30px;" |44
! style="width: 30px;" |45
! style="width: 30px;" |46
! style="width: 30px;" |47
! style="width: 30px;" |48
! style="width: 30px;" |49
! style="width: 30px;" |50
|-
!<math>\mu(n)</math>
|−1
|−1
|−1
|0
|0
|1
|−1
|0
|0
|0
|}

The first 50 values of the function are plotted below:
[[File:Moebius mu.svg|center|The 50 first values of <math>\mu(n)</math>]]

Larger values can be checked in: 
* [https://www.wolframalpha.com/input/?i=MoebiusMu+123 Wolframalpha]
* [https://oeis.org/A008683/b008683.txt the b-file of OEIS]

== Applications ==
=== Mathematical series ===
The [[Dirichlet series]] that [[Generating function|generates]] the Möbius function is the (multiplicative) inverse of the [[Riemann zeta function]]; if <math>s</math> is a complex number with real part larger than 1 we have

:<math>\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.</math>

This may be seen from its [[Euler product]]

:<math>\frac{1}{\zeta(s)} = \prod_{p \text{ prime}}{\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>

Also:

* <math>\sum\limits_{n=1}^{\infty} \frac{|\mu(n)|}{n^{s}} = \frac{\zeta(s)}{\zeta(2s)};</math>
* <math>\sum_{n=1}^\infty \frac{\mu(n)}{n}=0;</math>
* <math>\sum\limits_{n=1}^{\infty} \frac{\mu(n)\ln n}{n}=-1;</math>
* <math>\sum\limits_{n=1}^{\infty} \frac{\mu(n)\ln^{2} n}{n}=-2\gamma,</math> where <math>\gamma</math> is [[Euler's constant]].

The [[Lambert series]] for the Möbius function is

:<math>\sum_{n=1}^\infty \frac{\mu(n)q^n}{1-q^n} = q,</math>
which converges for <math>|q|<1</math>. For prime <math>\alpha\geq 2</math>, we also have

:<math>\sum_{n=1}^\infty \frac{\mu(\alpha n)q^n}{q^n-1} = \sum_{n \geq 0} q^{\alpha^n}, |q| < 1.</math>

=== Algebraic number theory ===
Gauss{{sfn|Gauss|1986|loc=Art. 81}} proved that for a prime number <math>p</math> the sum of its [[Primitive root modulo n#Arithmetic facts|primitive roots]] is congruent to <math>\mu(p-1) \mod p</math>.

If <math>\mathbb{F}_q</math> denotes the [[finite field]] of order <math>q</math> (where <math>q</math> is necessarily a prime power), then the number <math>N</math> of monic irreducible polynomials of degree <math>n</math> over <math>\mathbb{F}_q</math> is given by{{sfn|Jacobson|2009|loc=§4.13}}

:<math>N(q,n)=\frac{1}{n} \sum_{d\mid n} \mu(d)q^\frac{n}{d}.</math>

The Möbius function is used in the [[Möbius inversion formula]].

===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 <math>\log(p)</math>. Under [[second quantization]], multiparticle excitations are considered; these are given by <math>\log(n)</math> for any natural number <math>n</math>. 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 [[primon gas|primons]] are taken as [[boson]]s. If they are taken as [[fermion]]s, then the [[Pauli exclusion principle]] excludes squares. The operator 
[[(-1)^F|<math>(-1)^F</math>]] that distinguishes fermions and bosons is then none other than the Möbius function <math>\mu(n)</math>.

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]]'s attempted proof of the [[Riemann hypothesis]].{{sfn|Bost|Connes|1995|pp=411–457}}

== Properties ==
The Möbius function is [[multiplicative function|multiplicative]] (i.e., 
<math>\mu(ab)=\mu(a)\mu(b)</math> whenever <math>a</math> and <math>b</math> are [[coprime]]).
<blockquote>'''Proof''': Given two coprime numbers <math>m \geq n</math>, we induct on <math>mn</math>. If <math>mn = 1</math>, then <math>\mu(mn) = 1 = \mu(m) \mu(n)</math>. Otherwise, <math>m > n \geq 1</math>, so

:<math display="block">\begin{align}
0 &= \sum_{d | mn} \mu(d) \\
&= \mu(mn) +  \sum_{d | mn ; d < mn} \mu(d) \\
&\stackrel{\text{induction}}{=} \mu(mn) - \mu(m)\mu(n) + \sum_{d| m; d' | n} \mu(d)\mu(d') \\
&= \mu(mn) - \mu(m)\mu(n) + \sum_{d| m} \mu(d)\sum_{d' | n}\mu(d') \\
&= \mu(mn) - \mu(m)\mu(n) + 0
\end{align}
 </math>

</blockquote>
The sum of the Möbius function over all positive divisors of <math>n</math> (including <math>n</math> itself and 1) is zero except when <math>n=1</math>:

:<math>\sum_{d\mid n} \mu(d) = 
\begin{cases}
1 & \text{if } n=1, \\
0 & \text{if } n>1.
\end{cases}</math>

The equality above leads to the important [[Möbius inversion formula]] and is the main reason why <math>\mu</math> is of relevance in the theory of multiplicative and arithmetic functions.

Other applications of <math>\mu(n)</math> in combinatorics are connected with the use of the [[Pólya enumeration theorem]] in combinatorial groups and combinatorial enumerations.

There is a formula{{sfn|Hardy|Wright|1980| loc=(16.6.4), p. 239}} for calculating the Möbius function without directly knowing the factorization of its argument:

:<math>\mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,\,n)=1}} e^{2\pi i \frac{k}{n}},</math>

i.e. <math>\mu(n)</math> is the sum of the primitive <math>n</math>-th [[roots of unity]]. (However, the computational complexity of this definition is at least the same as that of the Euler product definition.)

Other identities satisfied by the Möbius function include

:<math>\sum_{k \leq n} \left\lfloor{ \frac{n}{k} }\right\rfloor \mu(k) = 1</math>

and

:<math>\sum_{jk \leq n} \sin\left( { \frac{\pi jk}{2}} \right) \mu(k) = 1</math>.

The first of these is a classical result while the second was published in 2020.{{sfn|Apostol|1976}}{{sfn|Kline|2020}} Similar identities hold for the [[Mertens function]].

=== Proof of the formula for the sum of <math>\mu</math> over divisors ===
The formula

:<math>\sum_{d \mid n} \mu(d)=
\begin{cases}
1 & \text{if } n=1, \\
0 & \text{if } n>1
\end{cases}</math>

can be written using [[Dirichlet convolution]] as:
<math>1 * \mu = \varepsilon</math>
where <math>\varepsilon</math> is the [[Dirichlet convolution#Properties|identity under the convolution]].

One way of proving this formula is by noting that the Dirichlet convolution of two [[multiplicative functions]] is again multiplicative. Thus it suffices to prove the formula for powers of primes. Indeed, for any prime 
<math>p</math> and for any <math>k>0</math>
:<math>1 * \mu (p^k) = \sum_{d \mid p^k} \mu(d)= \mu(1)+\mu(p)+\sum_{1<m<=k}\mu(p^m)=1-1+\sum0=0=\varepsilon(p^k)</math>,
while for <math>n=1</math>
:<math>1 * \mu (1) = \sum_{d \mid 1} \mu(d)= \mu(1) =1 =\varepsilon(1)</math>.

====Other proofs====
Another way of proving this formula is by using the identity

:<math>\mu(n) = \sum_{\stackrel{1\le k \le n }{\gcd(k,\,n)=1}} e^{2\pi i \frac{k}{n}},</math>

The formula above is then a consequence of the fact that the <math>n</math>th roots of unity sum to 0, since each <math>n</math>th root of unity is a primitive <math>d</math>th root of unity for exactly one divisor <math>d</math> of <math>n</math>.

However it is also possible to prove this identity from first principles. First note that it is trivially true when <math>n=1</math>. Suppose then that <math>n>1</math>. Then there is a bijection between the factors <math>d</math> of <math>n</math> for which <math>\mu(d)\neq 0</math> and the subsets of the set of all prime factors of <math>n</math>. The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets.

This last fact can be shown easily by induction on the cardinality <math>|S|</math> of a non-empty finite set <math>S</math>. First, if <math>|S|=1</math>, there is exactly one odd-cardinality subset of <math>S</math>, namely <math>S</math> itself, and exactly one even-cardinality subset, namely <math>\emptyset</math>. Next, if 
<math>|S|>1</math>, then divide the subsets of <math>S</math> into two subclasses depending on whether they contain or not some fixed element <math>x</math> in <math>S</math>. There is an obvious bijection between these two subclasses, pairing those subsets that have the same complement relative to the subset <math>\{x\}</math>. Also, one of these two subclasses consists of all the subsets of the set <math>S\setminus\{x\}</math>, and therefore, by the induction hypothesis, has an equal number of odd- and even-cardinality subsets. These subsets in turn correspond bijectively to the even- and odd-cardinality <math>\{x\}</math>-containing subsets of <math>S</math>. The inductive step follows directly from these two bijections.

A related result is that the binomial coefficients exhibit alternating entries of odd and even power which sum symmetrically.

===Average order===
The [[average order of an arithmetic function|mean value (in the sense of average orders)]] of the Möbius function is zero. This statement is, in fact, equivalent to the [[prime number theorem]].{{sfn|Apostol|1976|loc=§3.9}}

===<math>\mu(n)</math> sections===
<math>\mu(n)=0</math> [[if and only if]] <math>n</math> is divisible by the square of a prime. The first numbers with this property are

:4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, ... {{OEIS|id=A013929}}.

If <math>n</math> is prime, then <math>\mu(n)=-1</math>, but the converse is not true. The first non prime <math>n</math> for which <math>\mu(n)=-1</math> is <math> 30 = 2\times 3\times 5</math>. The first such numbers with three distinct prime factors ([[sphenic number]]s) are

:30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, ... {{OEIS|id=A007304}}.

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, ... {{OEIS|id=A046387}}.

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

:<math>M(n) = \sum_{k = 1}^n \mu(k)</math>

for every natural number {{mvar|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 <math>M(n)</math> and the [[Riemann hypothesis]].

From the formula

:<math>\mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,n)=1}} e^{2\pi i \frac{k}{n}},</math>

it follows that the Mertens function is given by

:<math>M(n)= -1+\sum_{a\in \mathcal{F}_n} e^{2\pi i a},</math>

where <math>\mathcal{F}_n</math> is the [[Farey sequence]] of order <math>n</math>.

This formula is used in the proof of the [[Farey sequence#Riemann hypothesis|Franel–Landau theorem]].{{sfn|Edwards|1974|loc=Ch. 12.2}}

==Generalizations==
===Incidence algebras===
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.

===Popovici's function===
Constantin Popovici{{sfn|Popovici|1963|pp=493–499}} defined a generalised Möbius function 
<math>\mu_k=\mu * \cdots * \mu</math> to be the <math>k</math>-fold [[Dirichlet convolution]] of the Möbius function with itself. It is thus again a multiplicative function with

:<math> \mu_k\left(p^a\right) = (-1)^a \binom{k}{a} \ </math>

where the binomial coefficient is taken to be zero if <math>a>k</math>. The definition may be extended to complex <math>k</math> by reading the binomial as a polynomial in <math>k</math>.{{sfn|Sándor|Crstici|2004|p=107}}

==Implementations==
* [https://functions.wolfram.com/NumberTheoryFunctions/MoebiusMu/ Mathematica]
* [https://maxima.sourceforge.io/docs/manual/maxima_singlepage.html#index-moebius Maxima]
* [https://www.geeksforgeeks.org/program-mobius-function/ geeksforgeeks] C++, Python3, Java, C#, PHP, JavaScript
* [https://rosettacode.org/wiki/M%C3%B6bius_function Rosetta Code]
* [https://doc.sagemath.org/html/en/reference/rings%20standard/sage/arith/misc.html?highlight=moebius Sage]

==See also==
{{Div col}}
* [[Liouville function]]
* [[Mertens function]]
* [[Ramanujan's sum]]
* [[Sphenic number]]
{{Div col end}}

==Notes==
{{notelist-lr}}

===Citations===
{{Reflist}}

==Sources==
{{refbegin|35em}}
* {{Cite book |last1=Abramowitz |first1=Milton |title=Handbook of mathematical functions: with formulas, graphs and mathematical tables [conference under the auspices of the National science foundation and the Massachusetts institute of technology] |last2=Stegun |first2=Irene A. |date=1972 |publisher=Dover |isbn=978-0-486-61272-0 |series=Dover books on advanced mathematics |location=New York |orig-year=1964}}
*{{Cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |title=Introduction to analytic number theory |publisher=Springer-Verlag |year=1976 |isbn=978-0-387-90163-3 |series=Undergraduate Texts in Mathematics |location=New York; Heidelberg |mr=0434929 |zbl=0335.10001}}
*{{Cite journal |last1=Bost |first1=J.-B. |last2=Connes |first2=Alain |year=1995 |title=Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory |url=https://cds.cern.ch/record/283504 |journal=Selecta Mathematica |series=New Series |volume=1 |issue=3 |pages=411–457 |doi=10.1007/BF01589495 |s2cid=116418599}}
*{{Cite journal |last1=Deléglise |first1=Marc |last2=Rivat |first2=Joël |year=1996 |title=Computing the summation of the Möbius function |url=https://projecteuclid.org/euclid.em/1047565447 |journal=Experimental Mathematics |volume=5 |issue=4 |pages=291–295 |doi=10.1080/10586458.1996.10504594 |s2cid=574146}}
*{{Cite book |last=Edwards |first=Harold |author-link=Harold Edwards (mathematician) |title=Riemann's Zeta Function |date=1974 |publisher=Dover Publications |isbn=0-486-41740-9 |location=Mineola, New York}}
*{{Cite book |last=Gauss |first=Carl Friedrich |author-link=Carl Friedrich Gauss |title=Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) |publisher=Chelsea |year=1965 |isbn=0-8284-0191-8 |edition=2nd |location=New York |translator-last=Maser |translator-first=H.}}
*{{Cite book |last=Gauss |first=Carl Friedrich |author-link=Carl Friedrich Gauss |title=Disquisitiones Arithemeticae |publisher=[[Springer Science+Business Media|Springer]] |year=1986 |isbn=0-387-96254-9 |edition=corrected 2nd |location=New York |translator-last=Clarke |translator-first=Arthur A.}}
*{{Cite book |last1=Hardy |first1=G. H. |author-link1=G. H. Hardy |url=https://archive.org/details/introductiontoth00hard |title=An Introduction to the Theory of Numbers |last2=Wright |first2=E. M. |author-link2=E. M. Wright |date=1980 |publisher=[[Oxford University Press]] |isbn=978-0-19-853171-5 |edition=5th |location=Oxford |url-access=registration |via=[[Internet Archive]] |orig-year=First edition published 1938 <!-- need to note this, otherwise readers will query how he managed to write this book, when he died in 1947 -->}}
*{{Cite journal |last=Kline |first=Jeffery |year=2020 |title=Unital Sums of the Möbius and Mertens Functions |url=https://cs.uwaterloo.ca/journals/JIS/VOL23/Kline/kline4.pdf |journal=Journal of Integer Sequences |volume=23 |issue=8 |pages=1–17}}
*{{Cite book |last=Jacobson |first=Nathan |author-link=Nathan Jacobson |title=Basic algebra I |publisher=Dover Publications |year=2009 |isbn=978-0-486-47189-1 |edition=2nd |orig-year=First published 1985}}
* {{springer| title = Möbius function
 | last = Klimov | first = N. I.
 | id = m/m064280
}}
*{{Cite journal |last=Möbius |first=A. F. |author-link=August Ferdinand Möbius |year=1832 |title=Über eine besondere Art von Umkehrung der Reihen |url=https://www.digizeitschriften.de/en/dms/img/?PID=GDZPPN002138654 |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=9 |pages=105–123}}
*{{Cite web |last=Pegg |first=Ed Jr |author-link=Ed Pegg Jr. |date=2003 |title=The Möbius function (and squarefree numbers) |url=http://www.mathpuzzle.com/MAA/02-Mobius%20Function/mathgames_11_03_03.html |website=Ed Pegg's Math Games |mode=cs2}}
*{{Cite journal |last=Popovici |first=Constantin P. |year=1963 |title=A generalization of the Möbius function |journal=Studii şi Cercetări Matematice |volume=14 |pages=493–499 |mr=0181602}}
*{{Cite book |last1=Sándor |first1=Jozsef |title=Handbook of number theory II |last2=Crstici |first2=Borislav |publisher=Kluwer Academic |year=2004 |isbn=1-4020-2546-7 |location=Dordrecht |zbl=1079.11001}}
*{{Cite book |title=Handbook of number theory I |publisher=[[Springer-Verlag]] |year=2006 |isbn=1-4020-4215-9 |editor-last=Sándor |editor-first=József |location=Dordrecht |pages=187–226 |zbl=1151.11300 |editor-last2=Mitrinović |editor-first2=Dragoslav S. |editor-last3=Crstici |editor-first3=Borislav}}
{{refend}}

==External links==
* {{mathworld|urlname= MoebiusFunction|title=Möbius function}}

{{DEFAULTSORT:Mobius Function}}
[[Category:Multiplicative functions]]