Jordan's totient function

For each positive integer $k$, Jordan's totient function $J\_k$ is multiplicative and may be evaluated as

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

• $\backslash sum\_\{d\; |\; n\; \}\; J\_k(d)\; =\; n^k.\; \backslash ,$

:which may be written in the language of Dirichlet convolutions asSándor & Crstici (2004) p.106

:: $J\_k(n)\; \backslash star\; 1\; =\; n^k\backslash ,$

:and via Möbius inversion as

::$J\_k(n)\; =\; \backslash mu(n)\; \backslash star\; n^k$.

:Since the Dirichlet generating function of $\backslash mu$ is $1/\backslash zeta(s)$ and the Dirichlet generating function of $n^k$ is $\backslash zeta(s-k)$, the series for $J\_k$ becomes

::$\backslash sum\_\{n\backslash ge\; 1\}\backslash frac\{J\_k(n)\}\{n^s\}\; =\; \backslash frac\{\backslash zeta(s-k)\}\{\backslash zeta(s)\}$.

• An average order of $J\_k(n)$ is

::$J\_k(n)\; \backslash sim\; \backslash frac\{n^k\}\{\backslash zeta(k+1)\}$.

• The Dedekind psi function is

::$\backslash psi(n)\; =\; \backslash frac\{J\_2(n)\}\{J\_1(n)\}$,

: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 $\backslash frac\{J\_k(n)\}\{J\_1(n)\}$ or $\backslash frac\{J\_\{2k\}(n)\}\{J\_k(n)\}$ can also be shown to be integer-valued multiplicative functions.

• $\backslash sum\_\{\backslash delta\backslash mid\; n\}\backslash delta^sJ\_r(\backslash delta)J\_s\backslash left(\backslash frac\{n\}\{\backslash delta\}\backslash right)\; =\; J\_\{r+s\}(n)$.Holden et al in external links. The formula is Gegenbauer's.

• The general linear group of matrices of order $m$ over $\backslash mathbf\{Z\}/n$ has orderAll of these formulas are from Andrica and Piticari in #External links.

:$$

|\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 $\backslash mathbf\{Z\}/n$ 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 $\backslash mathbf\{Z\}/n$ 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.

• Explicit lists in the OEIS are J₂ in {{OEIS2C|A007434}}, J₃ in {{OEIS2C|A059376}}, J₄ in {{OEIS2C|A059377}}, J₅ in {{OEIS2C|A059378}}, J₆ up to J₁₀ in {{OEIS2C|A069091}} up to {{OEIS2C|A069095}}.
• Multiplicative functions defined by ratios are J₂(n)/J₁(n) in {{OEIS2C|A001615}}, J₃(n)/J₁(n) in {{OEIS2C|A160889}}, J₄(n)/J₁(n) in {{OEIS2C|A160891}}, J₅(n)/J₁(n) in {{OEIS2C|A160893}}, J₆(n)/J₁(n) in {{OEIS2C|A160895}}, J₇(n)/J₁(n) in {{OEIS2C|A160897}}, J₈(n)/J₁(n) in {{OEIS2C|A160908}}, J₉(n)/J₁(n) in {{OEIS2C|A160953}}, J₁₀(n)/J₁(n) in {{OEIS2C|A160957}}, J₁₁(n)/J₁(n) in {{OEIS2C|A160960}}.
• Examples of the ratios J_2k(n)/J_k(n) are J₄(n)/J₂(n) in {{OEIS2C|A065958}}, J₆(n)/J₃(n) in {{OEIS2C|A065959}}, and J₈(n)/J₄(n) in {{OEIS2C|A065960}}.

{{reflist}}

• {{cite book | author=L. E. Dickson | author-link=Leonard Eugene Dickson | title=History of the Theory of Numbers, Vol. I | orig-year=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 | author-link=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 }}

• {{cite journal |author-link=Dorin Andrica |first1=Dorin |last1=Andrica |first2=Mihai |last2=Piticari

|url=https://eudml.org/doc/126410

|journal=Acta Universitatis Apulensis

|year=2004

|volume=7

|pages=13–22

|mr=2157944

|title=On some extensions of Jordan's arithmetic functions

}}

• {{cite web |first1=Matthew |last1=Holden |first2=Michael |last2=Orrison |first3=Michael |last3=Vrable |url=http://www.math.hmc.edu/~orrison/research/papers/totient.pdf |title=Yet Another Generalization of Euler's Totient Function |access-date=2011-12-21 |archive-url=https://web.archive.org/web/20160305132124/https://www.math.hmc.edu/~orrison/research/papers/totient.pdf |archive-date=2016-03-05 |url-status=dead }}

{{Totient}}

Category:Modular arithmetic

Category:Multiplicative functions