Jordan's totient function

In number theory, Jordan's totient function, denoted as ${\displaystyle J_{k}(n)}${\displaystyle J_{k}(n)}, where ${\displaystyle k}${\displaystyle k} is a positive integer, is a function of a positive integer, ${\displaystyle n}${\displaystyle n}, that equals the number of ${\displaystyle k}${\displaystyle k}-tuples of positive integers that are less than or equal to ${\displaystyle n}${\displaystyle n} and that together with ${\displaystyle n}${\displaystyle n} form a coprime set of ${\displaystyle k+1}${\displaystyle k+1} integers.

Jordan's totient function is a generalization of Euler's totient function, which is the same as ${\displaystyle J_{1}(n)}${\displaystyle J_{1}(n)}. The function is named after Camille Jordan.

For each positive integer ${\displaystyle k}${\displaystyle k}, Jordan's totient function ${\displaystyle J_{k}}${\displaystyle J_{k}} is multiplicative and may be evaluated as

${\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right),円}${\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right),円}, where ${\displaystyle p}${\displaystyle p} ranges through the prime divisors of ${\displaystyle n}${\displaystyle n}.

• ${\displaystyle \sum _{d|n}J_{k}(d)=n^{k}.,円}${\displaystyle \sum _{d|n}J_{k}(d)=n^{k}.,円}

which may be written in the language of Dirichlet convolutions as^[1]

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

and via Möbius inversion as

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

Since the Dirichlet generating function of ${\displaystyle \mu }${\displaystyle \mu } is ${\displaystyle 1/\zeta (s)}${\displaystyle 1/\zeta (s)} and the Dirichlet generating function of ${\displaystyle n^{k}}${\displaystyle n^{k}} is ${\displaystyle \zeta (s-k)}${\displaystyle \zeta (s-k)}, the series for ${\displaystyle J_{k}}${\displaystyle J_{k}} becomes

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

• An average order of ${\displaystyle J_{k}(n)}${\displaystyle J_{k}(n)} is

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

• The Dedekind psi function is

${\displaystyle \psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}}${\displaystyle \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 ${\displaystyle p^{-k}}${\displaystyle p^{-k}}), the arithmetic functions defined by ${\displaystyle {\frac {J_{k}(n)}{J_{1}(n)}}}${\displaystyle {\frac {J_{k}(n)}{J_{1}(n)}}} or ${\displaystyle {\frac {J_{2k}(n)}{J_{k}(n)}}}${\displaystyle {\frac {J_{2k}(n)}{J_{k}(n)}}} can also be shown to be integer-valued multiplicative functions.

• ${\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}${\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}.^[2]

• The general linear group of matrices of order ${\displaystyle m}${\displaystyle m} over ${\displaystyle \mathbf {Z} /n}${\displaystyle \mathbf {Z} /n} has order^[3]

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

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

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

The first two formulas were discovered by Jordan.

• Explicit lists in the OEIS are J₂ in OEIS: A007434 , J₃ in OEIS: A059376 , J₄ in OEIS: A059377 , J₅ in OEIS: A059378 , J₆ up to J₁₀ in OEIS: A069091 up to OEIS: A069095 .
• Multiplicative functions defined by ratios are J₂(n)/J₁(n) in OEIS: A001615 , J₃(n)/J₁(n) in OEIS: A160889 , J₄(n)/J₁(n) in OEIS: A160891 , J₅(n)/J₁(n) in OEIS: A160893 , J₆(n)/J₁(n) in OEIS: A160895 , J₇(n)/J₁(n) in OEIS: A160897 , J₈(n)/J₁(n) in OEIS: A160908 , J₉(n)/J₁(n) in OEIS: A160953 , J₁₀(n)/J₁(n) in OEIS: A160957 , J₁₁(n)/J₁(n) in OEIS: A160960 .
• Examples of the ratios J_2k(n)/J_k(n) are J₄(n)/J₂(n) in OEIS: A065958 , J₆(n)/J₃(n) in OEIS: A065959 , and J₈(n)/J₄(n) in OEIS: A065960 .

• L. E. Dickson (1971) [1919]. History of the Theory of Numbers, Vol. I . Chelsea Publishing. p. 147. ISBN 0-8284-0086-5. JFM 47.0100.04.
• M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. Vol. 206. Springer-Verlag. p. 11. ISBN 0-387-95143-1. Zbl 0971.11001.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.

• Andrica, Dorin; Piticari, Mihai (2004). "On some extensions of Jordan's arithmetic functions". Acta Universitatis Apulensis. 7: 13–22. MR 2157944.
• Holden, Matthew; Orrison, Michael; Vrable, Michael. "Yet Another Generalization of Euler's Totient Function" (PDF). Archived from the original (PDF) on 2016年03月05日. Retrieved 2011年12月21日.

• Modular arithmetic
• Multiplicative functions

• Articles with short description
• Short description is different from Wikidata