Named After: | August Ferdinand Möbius |
Publication Year: | 1832 |
Author: | August Ferdinand Möbius |
Terms Number: | infinite |
First Terms: | 1, −1, −1, 0, −1, 1, −1, 0, 0, 1 |
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
\mu(n)
\mu(x)
The Möbius function is defined by
The Möbius function can alternatively be represented as
\mu(n)=\delta\omega(n)\Omega(n)λ(n),
where
\delta
λ(n)
\omega(n)
n
\Omega(n)
n
Another characterization by Gauss is the sum of all primitive roots.[1]
The values of
\mu(n)
The first 50 values of the function are plotted below:
Larger values can be checked in:
The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if
s
infty | |
\sum | |
n=1 |
\mu(n) | = | |
ns |
1 | |
\zeta(s) |
.
This may be seen from its Euler product
1 | |
\zeta(s) |
=\prodp{\left(1-
1 | |
ps |
\right)}=\left(1-
1 | \right)\left(1- | |
2s |
1 | \right)\left(1- | |
3s |
1 | |
5s |
\right) …
Also:
infty | |
\sum\limits | |
n=1 |
|\mu(n)| | |
ns |
=
\zeta(s) | |
\zeta(2s) |
;
infty | |
\sum | |
n=1 |
\mu(n) | |
n |
=0;
infty | |
\sum\limits | |
n=1 |
\mu(n)lnn | |
n |
=-1;
infty | |
\sum\limits | |
n=1 |
\mu(n)ln2n | |
n |
=-2\gamma,
\gamma
The Lambert series for the Möbius function is
infty | |
\sum | |
n=1 |
\mu(n)qn | |
1-qn |
=q,
|q|<1
\alpha\geq2
infty | |
\sum | |
n=1 |
\mu(\alphan)qn | |
qn-1 |
=\sumn
\alphan | |
q |
,|q|<1.
Gauss proved that for a prime number
p
\mu(p-1) (\modp)
If
Fq
q
q
N
n
Fq
N(q,n)= | 1 |
n |
\sumd\mid
| ||||
\mu(d)q |
.
The Möbius function is used in the Möbius inversion formula.
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)
log(n)
n
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
\mu(n)
The free Riemann gas has a number of other interesting connections to number theory, including the fact that the partition function is the Riemann zeta function. This idea underlies Alain Connes's attempted proof of the Riemann hypothesis.
The Möbius function is multiplicative (i.e.,
\mu(ab)=\mu(a)\mu(b)
a
b
Proof: Given two coprime numbersThe sum of the Möbius function over all positive divisors of, we induct onm\geqn
. Ifmn
, thenmn=1
. Otherwise,\mu(mn)=1=\mu(m)\mu(n)
, som>n\geq1
n
n
n=1
\sumd\mid\mu(d)=\begin{cases} 1&ifn=1,\\ 0&ifn>1. \end{cases}
The equality above leads to the important Möbius inversion formula and is the main reason why
\mu
Other applications of
\mu(n)
There is a formula for calculating the Möbius function without directly knowing the factorization of its argument:
\mu(n)=\sum\stackrel{1\le{\gcd(k,n)=1}}
| ||||||
e |
,
i.e.
\mu(n)
n
Other identities satisfied by the Möbius function include
\sumk\left\lfloor{
n | |
k |
}\right\rfloor\mu(k)=1
and
\sumjk\sin\left({
\pijk | |
2 |
The first of these is a classical result while the second was published in 2020. Similar identities hold for the Mertens function.
\mu
\sumd\mu(d)= \begin{cases} 1&ifn=1,\\ 0&ifn>1 \end{cases}
can be written using Dirichlet convolution as:
1*\mu=\varepsilon
\varepsilon
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
p
k>0
1*\mu(pk)=
\sum | |
d\midpk |
\mu(d)=\mu(1)+\mu(p)+\sum1<m<=k\mu(pm)=1-1+\sum0=0=\varepsilon(pk)
n=1
1*\mu(1)=\sumd\mu(d)=\mu(1)=1=\varepsilon(1)
Another way of proving this formula is by using the identity
\mu(n)=\sum\stackrel{1\le{\gcd(k,n)=1}}
| ||||||
e |
,
The formula above is then a consequence of the fact that the
n
n
d
d
n
However it is also possible to prove this identity from first principles. First note that it is trivially true when
n=1
n>1
d
n
\mu(d) ≠ 0
n
This last fact can be shown easily by induction on the cardinality
|S|
S
|S|=1
S
S
\emptyset
|S|>1
S
x
S
\{x\}
S\setminus\{x\}
\{x\}
S
A related result is that the binomial coefficients exhibit alternating entries of odd and even power which sum symmetrically.
The 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.
\mu(n)
\mu(n)=0
n
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
\mu(n)=-1
n
\mu(n)=-1
30=2 x 3 x 5
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, ... .
In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by
M(n)=
n | |
\sum | |
k=1 |
\mu(k)
for every natural number . 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)
From the formula
\mu(n)=\sum\stackrel{1\le{\gcd(k,n)=1}}
| ||||||
e |
,
it follows that the Mertens function is given by
M(n)=-1+\suma\inn}e2\pi,
where
l{F}n
n
This formula is used in the proof of the Franel–Landau theorem.
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 divisibility. See the article on incidence algebras for the precise definition and several examples of these general Möbius functions.
Constantin Popovici defined a generalised Möbius function
\muk=\mu* … *\mu
k
a\right) | |
\mu | |
k\left(p |
=(-1)a\binom{k}{a}
where the binomial coefficient is taken to be zero if
a>k
k
k