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Let ''k'' be a [[positive integer]]. In [[number theory]], '''Jordan's totient function''' <math>J_k(n)</math> of a positive integer ''n'' is the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a [[coprime]] (''k''&nbsp;+&nbsp;1)-tuple together with ''n''.  This is a generalisation of Euler's [[totient function]], which is ''J''<sub>1</sub>. The function is named after [[Camille Jordan]].
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==Definition==
 
Jordan's totient function is [[multiplicative function|multiplicative]] and may be evaluated as
:<math>J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) .\,</math>
 
==Properties==
 
* <math>\sum_{d | n } J_k(d) = n^k. \, </math>
which may be written in the language of [[Dirichlet convolution]]s as<ref>Sándor & Crstici (2004) p.106</ref>
: <math>J_k(n) \star 1 = n^k\,</math>
and via [[Möbius inversion formula|Möbius inversion]] as
:<math>J_k(n) = \mu(n) \star n^k</math>.
Since the [[Dirichlet generating function]] of μ is 1/ζ(s) and the
Dirichlet generating function of n<sup>k</sup> is ζ(s-k), the series for
J<sub>k</sub> becomes
:<math>\sum_{n\ge 1}\frac{J_k(n)}{n^s} = \frac{\zeta(s-k)}{\zeta(s)}</math>.
 
* An [[Average order of an arithmetic function|average order]] of ''J''<sub>''k''</sub>(''n'') is 
:<math>\frac{n^k}{\zeta(k+1)}</math>.
* The [[Dedekind psi function]] is
:<math>\psi(n) = \frac{J_2(n)}{J_1(n)}</math>,
and by inspection of the definition (recognizing that each factor in the product
over the primes is a cyclotomic polynomial of p<sup>-k</sup>), the arithmetic functions
defined by <math>\frac{J_k(n)}{J_1(n)}</math> or <math>\frac{J_{2k}(n)}{J_k(n)}</math>
can also be shown to be integer-valued multiplicative functions.
 
* <math>
\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)
</math> &nbsp; &nbsp; &nbsp;<ref>Holden et al in external links The formula is Gegenbauer's</ref>
 
==Order of matrix groups==
 
The [[general linear group]] of matrices of order ''m'' over '''Z'''<sub>''n''</sub> has order<ref>All of these formulas are from Andrici and Priticari in [[#External links]]</ref>
:<math>
|\operatorname{GL}(m,\mathbf{Z}_n)|=n^{\frac{m(m-1)}{2}}\prod_{k=1}^m J_k(n).
</math>
 
The [[special linear group]] of matrices of order ''m'' over ''Z''<sub>''n''</sub> has order
:<math>
|\operatorname{SL}(m,\mathbf{Z}_n)|=n^{\frac{m(m-1)}{2}}\prod_{k=2}^m J_k(n).
</math>
 
The [[symplectic group]] of matrices of order ''m'' over ''Z''<sub>''n''</sub> has order
:<math>
|\operatorname{Sp}(2m,\mathbf{Z}_n)|=n^{m^2}\prod_{k=1}^m J_{2k}(n).
</math>
 
The first two formulas were discovered by Jordan.
 
==Examples==
 
Explicit lists in the [[OEIS]] are
J<sub>2</sub> in {{OEIS2C|A007434}},
J<sub>3</sub> in {{OEIS2C|A059376}},
J<sub>4</sub> in {{OEIS2C|A059377}},
J<sub>5</sub> in {{OEIS2C|A059378}},
J<sub>6</sub> up to J<sub>10</sub> in {{OEIS2C|A069091}}
up to {{OEIS2C|A069095}}.
       
                       
Multiplicative functions defined by ratios are
J<sub>2</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A001615}},
J<sub>3</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160889}},
J<sub>4</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160891}},
J<sub>5</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160893}},
J<sub>6</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160895}},
J<sub>7</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160897}},
J<sub>8</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160908}},
J<sub>9</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160953}},
J<sub>10</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160957}},
J<sub>11</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160960}}.
       
                       
Examples of the ratios J<sub>2k</sub>(n)/J<sub>k</sub>(n) are
J<sub>4</sub>(n)/J<sub>2</sub>(n) in {{OEIS2C|A065958}},
J<sub>6</sub>(n)/J<sub>3</sub>(n) in {{OEIS2C|A065959}},
and
J<sub>8</sub>(n)/J<sub>4</sub>(n) in {{OEIS2C|A065960}}.
 
==Notes==
 
{{reflist}}
 
==References==
 
* {{cite book | author=L. E. Dickson | authorlink=Leonard Eugene Dickson | title=[[History of the Theory of Numbers]], Vol. I | origyear=1919 |year=1971 | publisher=[[Chelsea Publishing]] | isbn=0-8284-0086-5 | jfm=47.0100.04 | page=147 }}
*{{cite book | title=Problems in Analytic Number Theory | author=M. Ram Murty | authorlink=M. Ram Murty  | volume=206 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | year=2001 | isbn=0-387-95143-1 | zbl=0971.11001  | page=11 }}
* {{cite book | 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 | pages=32–36 | zbl=1079.11001 }}
 
==External links==
*{{cite journal|first1=Dorin |last1=Andrica |first2= Mihai |last2=Piticari
|url=http://www.emis.de/journals/AUA/acta7/Andrica%20.pdf
|journal=Acta universitatis Apulensis
|year=2004
|number=7
|mr=2157944
|title= On some Extensions of Jordan's arithmetical Functions
}}
* {{cite web |first1=Matthew |last1=Holden|first2=Michael |last2=Orrison
|first3=Michael |last3=Varble
|url=http://www.math.hmc.edu/~orrison/research/papers/totient.pdf
|title= Yet another Generalization of Euler's Totient Function
}}
 
{{Totient}}
 
[[Category:Number theory]]
[[Category:Modular arithmetic]]
[[Category:Multiplicative functions]]

Latest revision as of 17:20, 6 December 2014

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