Porosity

Let k be a positive integer. In number theory, Jordan's totient function $J_{k}(n)$ 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 + 1)-tuple together with n. This is a generalisation of Euler's totient function, which is J₁. The function is named after Camille Jordan.

Definition

Jordan's totient function is multiplicative and may be evaluated as

$J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right).\,$

Properties

• $\sum _{d|n}J_{k}(d)=n^{k}.\,$

which may be written in the language of Dirichlet convolutions as

$J_{k}(n)\star 1=n^{k}\,$

and via Möbius inversion as

$J_{k}(n)=\mu (n)\star n^{k}$.

Since the Dirichlet generating function of μ is 1/ζ(s) and the Dirichlet generating function of n^k is ζ(s-k), the series for J_k becomes

$\sum _{n\geq 1}{\frac {J_{k}(n)}{n^{s}}}={\frac {\zeta (s-k)}{\zeta (s)}}$.

• An average order of J_k(n) is

${\frac {n^{k}}{\zeta (k+1)}}$.

• The Dedekind psi function is

$\psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}$,

and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of p^-k), the arithmetic functions defined by ${\frac {J_{k}(n)}{J_{1}(n)}}$ or ${\frac {J_{2k}(n)}{J_{k}(n)}}$ can also be shown to be integer-valued multiplicative functions.

• $\sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)$      

Order of matrix groups

The general linear group of matrices of order m over Z_n has order

$|\operatorname {GL} (m,\mathbf {Z} _{n})|=n^{\frac {m(m-1)}{2}}\prod _{k=1}^{m}J_{k}(n).$

The special linear group of matrices of order m over Z_n has order

$|\operatorname {SL} (m,\mathbf {Z} _{n})|=n^{\frac {m(m-1)}{2}}\prod _{k=2}^{m}J_{k}(n).$

The symplectic group of matrices of order m over Z_n has order

$|\operatorname {Sp} (2m,\mathbf {Z} _{n})|=n^{m^{2}}\prod _{k=1}^{m}J_{2k}(n).$

The first two formulas were discovered by Jordan.

Examples

Explicit lists in the OEIS are J₂ in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₃ in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₄ in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₅ in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₆ up to J₁₀ in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. up to Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

Multiplicative functions defined by ratios are J₂(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₃(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₄(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₅(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₆(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₇(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₈(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₉(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₁₀(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₁₁(n)/J₁(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

Examples of the ratios J_2k(n)/J_k(n) are J₄(n)/J₂(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., J₆(n)/J₃(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., and J₈(n)/J₄(n) in Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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