Let
be a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
. In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, the Jordan's totient function
of a positive integer
equals the number of
-
tuples of positive integers that are less than or equal to
and that together with
form a
coprime set
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
of
integers.
Jordan's totient function is a generalization of Euler's
totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
, which is given by
. The function is named after
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
.
Definition
For each
, Jordan's totient function
is
multiplicative and may be evaluated as
:
, where
ranges through the prime divisors of
.
Properties
*
:which may be written in the language of
Dirichlet convolutions as
::
:and via
Möbius inversion as
::
.
:Since the
Dirichlet generating function of
is
and the Dirichlet generating function of
is
, the series for
becomes
::
.
* An
average order of
is
::
.
* The
Dedekind psi function
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by
: \psi(n) = n \prod_\left(1+\frac\right),
where the product is taken over all primes p dividing n. (By convention, \psi(1), which is ...
is
::
,
:and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of
), the arithmetic functions defined by
or
can also be shown to be integer-valued multiplicative functions.
*
.
Order of matrix groups
* The
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of matrices of order
over
has order
[All of these formulas are from Andrici and Priticari in #External links]
:
* The
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
of matrices of order
over
has order
:
* The
symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
of matrices of order
over
has order
:
The first two formulas were discovered by Jordan.
Examples
* Explicit lists in the
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
are J
2 in , J
3 in , J
4 in , J
5 in , J
6 up to J
10 in up to .
* Multiplicative functions defined by ratios are J
2(n)/J
1(n) in , J
3(n)/J
1(n) in , J
4(n)/J
1(n) in , J
5(n)/J
1(n) in , J
6(n)/J
1(n) in , J
7(n)/J
1(n) in , J
8(n)/J
1(n) in , J
9(n)/J
1(n) in , J
10(n)/J
1(n) in , J
11(n)/J
1(n) in .
* Examples of the ratios J
2k(n)/J
k(n) are J
4(n)/J
2(n) in , J
6(n)/J
3(n) in , and J
8(n)/J
4(n) in .
Notes
References
*
*
*
External links
*
*
{{Totient
Modular arithmetic
Multiplicative functions