Zhordanov totiyent

Let be $k$ ${\ displaystyle k}$ $k$ Is a natural number . Jordan totiant or Jordan Function ^[1] $J_{k}(n)$ ${\ displaystyle J_ {k} (n)}$ ${\ displaystyle J_ {k} (n)}$ positive integer $n$ ${\ displaystyle n}$ $n$ Is a number $k$ ${\ displaystyle k}$ $k$ tuples of positive integers less than or equal to $n$ ${\ displaystyle n}$ $n$ forming $(k+1)$ ${\ displaystyle (k + 1)}$ $(k + 1)$ tuple whose greatest common divisor is mutually simple with $n$ ${\ displaystyle n}$ $n$ . The function is a generalization of the Euler function , which is equal to $J_{1}$ ${\ displaystyle J_ {1}}$ $J_ {1}$ . The function bears the name of the French mathematician Jordan .

Content

Definition

The Jordan function is multiplicative and can be calculated by the formula

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

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

where

p

{\ displaystyle p}

runs through simple divisors

n

{\ displaystyle n}

.

Properties

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

what can be written in the language of Dirichlet bundles as ^[2]

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

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

,

and through Möbius appeals as

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

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

.

Since the generating Dirichlet function $\mu$ ${\ displaystyle \ mu}$ is equal to $1/\zeta (s)$ ${\ displaystyle 1 / \ zeta (s)}$ , and the generating Dirichlet function $n^{k}$ ${\ displaystyle n ^ {k}}$ is equal to $\zeta (s-k)$ ${\ displaystyle \ zeta (sk)}$ row for $J_{k}$ ${\ displaystyle J_ {k}}$ turns into

\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 (sk)} {\ zeta (s)} }}

.

for $J_{k}(n)$ ${\ displaystyle J_ {k} (n)}$ is equal to

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

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

.

Dedekind's psi function is

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

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

,

and in the study of the definition (we note that each factor in the product of primes is a circular polynomial $p_{-k}$ ${\ displaystyle p _ {- k}}$ ), it can be shown that arithmetic functions defined as ${\frac {J_{k}(n)}{J_{1}(n)}}$ ${\ displaystyle {\ frac {J_ {k} (n)} {J_ {1} (n)}}}$ or ${\frac {J_{2k}(n)}{J_{k}(n)}}$ ${\ displaystyle {\ frac {J_ {2k} (n)} {J_ {k} (n)}}}$ , are integer multiplicative functions.

$\sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)$ ${\ displaystyle \ sum _ {\ delta \ mid n} \ delta ^ {s} J_ {r} (\ delta) J_ {s} \ left ({\ frac {n} {\ delta}} \ right) = J_ {r + s} (n)}$ . ^[3] ^[4]

Matrix Group Order

Complete linear group of order matrices $m$ ${\ displaystyle m}$ above $\mathbb {Z} _{n}$ ${\ displaystyle \ mathbb {Z} _ {n}}$ has the order ^[5]

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

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

Special linear order group $m$ ${\ displaystyle m}$ above $\mathbb {Z} _{n}$ ${\ displaystyle \ mathbb {Z} _ {n}}$ has order

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

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

Symplectic group of order matrices $m$ ${\ displaystyle m}$ above $\mathbb {Z} _{n}$ ${\ displaystyle \ mathbb {Z} _ {n}}$ has order

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

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

The first two formulas were discovered by Jordan.

Examples

Lists in OEIS J ₂ on A007434 , J ₃ on A059376 , J ₄ on A059377 , J ₅ on A059378 , J ₆ to J ₁₀ on lists A069091 - A069095 .

Multiplicative functions defined by the relation J ₂ (n) / J ₁ (n) in A001615 , J ₃ (n) / J ₁ (n) in A160889 , J ₄ (n) / J ₁ (n) in A160891 , J ₅ ( n) / J ₁ (n) in A160893 , J ₆ (n) / J ₁ (n) in A160895 , J ₇ (n) / J ₁ (n) in A160897 , J ₈ (n) / J ₁ (n) in A160908 , J ₉ (n) / J ₁ (n) in A160953 , J ₁₀ (n) / J ₁ (n) in A160957 , J ₁₁ (n) / J ₁ (n) in A160960 .

Examples of relations J _2k (n) / J _k (n): J ₄ (n) / J ₂ (n) in A065958 , J ₆ (n) / J ₃ (n) in A065959 and J ₈ (n) / J ₄ (n) in A065960 .

Notes

↑ There are other functions of Jordan. So, Merzlyakov writes: “ Theorem . There is a “Jordan Function” $J:\mathbb {N} \to \mathbb {N}$ ${\ displaystyle J: \ mathbb {N} \ to \ mathbb {N}}$ with the following property: every finite group G from $GL_{n}(\mathbb {C} )$ ${\ displaystyle GL_ {n} (\ mathbb {C})}$ contains an abelian normal subgroup A with index $\leqslant L(n)$ ${\ displaystyle \ leqslant L (n)}$ . "
↑ Sándor, Crstici, 2004 , p. 106.
↑ Holden, Orrison, Varble .
↑ Gegenbauer formula
↑ Andrica, Piticari, 2004 .

Literature

Dickson LE History of the Theory of Numbers , Vol. I. - Chelsea Publishing , 1971. - S. 147. - ISBN 0-8284-0086-5 .
M. Ram Murty. Problems in Analytic Number Theory. - Springer-Verlag , 2001. - T. 206. - S. 11. - ( Graduate Texts in Mathematics ). - ISBN 0-387-95143-1 .
Jozsef Sándor, Borislav Crstici. Handbook of number theory II. - Dordrecht: Kluwer Academic, 2004. - S. 32–36. - ISBN 1-4020-2546-7 .

Matthew Holden, Michael Orrison, Michael Varble. Yet another Generalization of Euler's Totient Function .
Dorin Andrica, Mihai Piticari. On some Extensions of Jordan's arithmetical Functions // Acta universitatis Apulensis. - 2004. - No. 7 .

Links

Merzlyakov Yu.I. Rational groups. - Moscow: "Science", 1980. - (Modern Algebra).

[1] There are other functions of Jordan. So, Merzlyakov writes: “ Theorem . There is a “Jordan Function” $J:\mathbb {N} \to \mathbb {N}$ ${\ displaystyle J: \ mathbb {N} \ to \ mathbb {N}}$ with the following property: every finite group G from $GL_{n}(\mathbb {C} )$ ${\ displaystyle GL_ {n} (\ mathbb {C})}$ contains an abelian normal subgroup A with index $\leqslant L(n)$ ${\ displaystyle \ leqslant L (n)}$ . "

[_c08aaedb4c35fc8b-2] Sándor, Crstici, 2004 , p. 106.

[_83ee239175fcc42b-3] Holden, Orrison, Varble .

[4] Gegenbauer formula

[_9d5bc07da285eef4-5] Andrica, Piticari, 2004 .