(Redirected from Mobius function)
![Pending changes are displayed on this page Pending changes are displayed on this page](/w/extensions/FlaggedRevs/frontend/modules/img/1.png)
There are no approved revisions of this page, so it may
not have been
reviewed.
The
Möbius function, named after
August Ferdinand Möbius (where Möbius is sometimes transliterated, without the
diacritic, as Mœbius,
[1] Moebius or Mobius), denoted
, tells whether a
positive integer is
squarefree, and if so, whether it has an
odd or
even number of
prime factors. Thus
![{\displaystyle \mu (n):={\begin{cases}-1&{\text{ if }}\Omega (n)=\omega (n){\text{ and }}\omega (n)\equiv 1{\pmod {2}},\\\;\;\;0&{\text{ if }}\Omega (n)>\omega (n),\\\;\;\;1&{\text{ if }}\Omega (n)=\omega (n){\text{ and }}\omega (n)\equiv 0{\pmod {2}},\end{cases}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8849accc03b7ee1ed173b2e889b9b6db2d80b672)
where
is the
number of distinct prime factors of
,
is the
number of prime factors (with repetition) of
. Note that
is
squarefree if and only if
.
Exploiting the fact that the value of
alternates between
+ 1 and
− 1, depending on the parity of
(see
A033999 for
), we can condense the definition into
![{\displaystyle \mu (n):={\begin{cases}(-1)^{\omega (n)}&{\text{ if }}\Omega (n)=\omega (n),\\0&{\text{ if }}\Omega (n)>\omega (n),\end{cases}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e579d969ac2aeb4032f47ede1bdd46cae34f1d45)
and using the Iverson bracket, we can further condense the definition into
![{\displaystyle \mu (n):=[\omega (n)=\Omega (n)](-1)^{\omega (n)}.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7b744e3d47b8da5b7852139176bc768f18bd376a)
(When
, the Iverson bracket is
0 and there is no need to evaluate
, this being referred to as
short-circuit evaluation.)
As we no longer consider
1 a prime number (as was the case in August Ferdinand Möbius’s time
[2]), the number
1 has an even number of prime factors, namely zero, and therefore
. This simplifies certain identities as we shall see later on. For more values of the Möbius function, see
A008683.
Table of Möbius function related values
Möbius function related values
|
|
Möbius
A008683
: A030059
: A013929
: A030229
|
Mertens
A002321
|
1
|
0, 0
|
1
|
1
|
2
|
1, 1
|
–1
|
0
|
3
|
1, 1
|
–1
|
–1
|
4
|
2, 1
|
0
|
–1
|
5
|
1, 1
|
–1
|
–2
|
6
|
2, 2
|
1
|
–1
|
7
|
1, 1
|
–1
|
–2
|
8
|
3, 1
|
0
|
–2
|
9
|
2, 1
|
0
|
–2
|
10
|
2, 2
|
1
|
–1
|
11
|
1, 1
|
–1
|
–2
|
12
|
3, 2
|
0
|
–2
|
13
|
1, 1
|
–1
|
–3
|
14
|
2, 2
|
1
|
–2
|
15
|
2, 2
|
1
|
–1
|
16
|
4, 1
|
0
|
–1
|
17
|
1, 1
|
–1
|
–2
|
18
|
3, 2
|
0
|
–2
|
19
|
1, 1
|
–1
|
–3
|
20
|
3, 2
|
0
|
–3
|
21
|
2, 2
|
1
|
–2
|
22
|
2, 2
|
1
|
–1
|
23
|
1, 1
|
–1
|
–2
|
24
|
4, 2
|
0
|
–2
|
25
|
2, 1
|
0
|
–2
|
26
|
2, 2
|
1
|
–1
|
27
|
3, 1
|
0
|
–1
|
28
|
3, 2
|
0
|
–1
|
29
|
1, 1
|
–1
|
–2
|
30
|
3, 3
|
–1
|
–3
|
31
|
1, 1
|
–1
|
–4
|
32
|
5, 1
|
0
|
–4
|
The value is available in PARI/GP as “moebius(n)” and as “MoebiusMu[n]” in Mathematica.
Asymptotic behavior
The summatory quadratfrei function is defined as
![{\displaystyle Q(n):=\sum _{i=1}^{n}|\mu (n)|,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5dc58337b11ef88e38b776d2aad1b7a20b90104f)
where
is the
quadratfrei function, the
characteristic function of squarefree numbers.
The asymptotic density of squarefree numbers corresponds to the probability that two randomly chosen integers are coprime
![{\displaystyle \lim _{n\to \infty }{\frac {Q(n)}{n}}=\prod _{n=1}^{\infty }{\bigg (}1-{\frac {1}{{p_{n}}^{2}}}{\bigg )}={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}},}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b490740a0bfa06f06fd5f67daba3d872e63f7b5a)
where
is the
th prime number, and
is the
Riemann zeta function.
The asymptotic density of squarefree numbers with an odd number of prime factors is equal to the asymptotic density of squarefree numbers with an even number of prime factors, i.e.
![{\displaystyle \lim _{n\to \infty }{\frac {\sum _{i=1}^{n}[\mu (i)=-1]}{n}}=\lim _{n\to \infty }{\frac {\sum _{i=1}^{n}[\mu (i)=+1]}{n}}={\frac {1}{2\zeta (2)}}={\frac {3}{\pi ^{2}}},}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/29511d8773c9172b84da974bf72204dbc4ba91e0)
where
is the
Iverson bracket.
The graph of the Mertens function (the Mertens function being the summatory Möbius function) seems to indicate an average negative bias for the Mertens function, which would mean that there is a bias (eerily similar to the Chebyshev bias) in favor of the squarefree numbers with an odd number of prime factors over the squarefree numbers with an even number of prime factors. The existence or not of such a bias, if small enough, would have no effect on the asymptotic behavior.
Partial sums of the Möbius function
The partial sums of the Möbius function give the Mertens function
![{\displaystyle M(n):=\sum _{i=1}^{n}\mu (i).}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f8f281aee2fe965d5a77d2cc0b124184a50ecc74)
With the exception of
, we find that, given the set of divisors of
,
, where
is the
number of divisors of
, then
[3]
![{\displaystyle \sum _{i=1}^{\tau (n)}\mu (d_{i})=0.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7fa70694030e36c8f812d3cbeb78a111db45413a)
Dirichlet generating function
Since the inverse of the Riemann zeta function is given by the Dirichlet series with the Möbius function as Dirichlet character (Dirichlet generating sequence)
![{\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}},\ \Re (s)>1,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/bffa6ff276f0f63adfb2179b7df0db7db2168d02)
we have that the Dirichlet generating function of the Möbius function is the inverse of the Riemann zeta function
![{\displaystyle D_{\{\mu (n)\}}(s)={\frac {1}{\zeta (s)}}.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8982d591809207110b59383927bb24f5230cdd48)
The Möbius function is the sum of the number of points on multidimensional hyperboloids:
![{\displaystyle \mu (n)={\underset {n=1}{1}}-{\underset {a=n}{\sum _{a\geq 2}}}1+{\underset {ab=n}{\sum _{a\geq 2}\sum _{b\geq 2}}}1-{\underset {abc=n}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}}}1+{\underset {abcd=n}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}\sum _{d\geq 2}}}1-\cdots }](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/06bb7c766efcb0fdfac3e90f9c320f773702f788)
Properties
![{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}=\prod _{p{\rm {~prime}}}{\bigg (}1-{\frac {1}{p^{s}}}{\bigg )}=\prod _{n=2}^{\infty }{\bigg (}1-{\frac {1}{n^{s}}}{\bigg )}^{\chi _{\rm {prime}}(n)},\,{\mathfrak {R}}(s)>1,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2ddb579e2359e7461d8e6e858cc02ff9b4840cf0)
where
is the
Riemann zeta function and
is the
characteristic function of prime numbers.
![{\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)\log(n)}{n}}=-1,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0f39d33c22e05f8d675b33c9c7f569b31290d771)
where
is the
quadratfrei function.
![{\displaystyle \prod _{n=1}^{\infty }(1-x^{n})^{\frac {\mu (n)}{n}}=e^{-x},\,|x|<1.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d1005b581c6c0c81cffc3a801025f27f50b4562c)
Möbius inversion
The Möbius function is used to define the Möbius transform (or Möbius inversion) of a sequence
![{\displaystyle b_{n}=\sum _{d|n}\mu {\big (}{\tfrac {n}{d}}{\big )}a_{d},\ a_{n}=\sum _{d|n}b_{d}.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/40b9099d82cbab0d64966369284557d865895cb1)
Sequences
A030059 : Numbers that are the product of an odd number of distinct primes.
-
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, ...}
A013929 : Numbers that are not squarefree. Numbers that are divisible by a square greater than
1. The complement of
A005117.
-
{4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, ...}
A030229 : Numbers that are the product of an even number of distinct primes.
-
{1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, ...}
A008683 Moebius (or Mobius) function
.
-
{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, 1, 0, 0, –1, –1, –1, 0, 1, 1, 1, 0, –1, 1, 1, 0, –1, –1, –1, 0, 0, 1, –1, 0, 0, 0, 1, 0, –1, 0, 1, 0, 1, 1, –1, 0, –1, 1, 0, 0, 1, –1, –1, ...}
A002321 Mertens function:
, where
is Möbius function (
A008683).
-
{1, 0, –1, –1, –2, –1, –2, –2, –2, –1, –2, –2, –3, –2, –1, –1, –2, –2, –3, –3, –2, –1, –2, –2, –2, –1, –1, –1, –2, –3, –4, –4, –3, –2, –1, –1, –2, –1, 0, 0, –1, –2, –3, –3, –3, –2, –3, –3, –3, –3, –2, –2, –3, –3, ...}
See also
Notes