Jordan's totient function

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

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

Definition

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

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

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 $\mu$ is $1/\zeta (s)$ and the Dirichlet generating function of $n^{k}$ is $\zeta (s-k)$, 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

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

• 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 $\mathbf {Z} /n$ 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 $\mathbf {Z} /n$ 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 $\mathbf {Z} /n$ 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 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.

References

• 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.

External links

• 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 2016-03-05. Retrieved 2011-12-21.