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Jordan's totient function

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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 J 1 . The function is named after Camille Jordan.

Contents

Definition

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

J k ( n ) = n k ∏ p | n ( 1 − 1 p k ) .

Properties

  • ∑ d | n J k ( d ) = n k .

    • which may be written in the language of Dirichlet convolutions as

    J k ( n ) ⋆ 1 = n k

    and via Möbius inversion as

    J k ( n ) = μ ( n ) ⋆ 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

    ∑ n ≥ 1 J k ( n ) n s = ζ ( s − k ) ζ ( s ) .
  • An average order of J k ( n ) is
    • n k ζ ( k + 1 ) .
  • The Dedekind psi function is
    • ψ ( n ) = 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 J k ( n ) J 1 ( n ) or J 2 k ( n ) J k ( n ) can also be shown to be integer-valued multiplicative functions.

  • ∑ δ ∣ n δ s J r ( δ ) J s ( n δ ) = J r + s ( n ) .      

    • Order of matrix groups

    The general linear group of matrices of order m over Z n has order

    | GL ⁡ ( m , Z n ) | = n m ( m − 1 ) 2 ∏ k = 1 m J k ( n ) .

    The special linear group of matrices of order m over Z n has order

    | SL ⁡ ( m , Z n ) | = n m ( m − 1 ) 2 ∏ k = 2 m J k ( n ) .

    The symplectic group of matrices of order m over Z n has order

    | Sp ⁡ ( 2 m , Z n ) | = n m 2 ∏ k = 1 m J 2 k ( n ) .

    The first two formulas were discovered by Jordan.

    Examples

    Explicit lists in the OEIS are J2 in  A007434, J3 in  A059376, J4 in  A059377, J5 in  A059378, J6 up to J10 in  A069091 up to  A069095.


    Multiplicative functions defined by ratios are J2(n)/J1(n) in  A001615, J3(n)/J1(n) in  A160889, J4(n)/J1(n) in  A160891, J5(n)/J1(n) in  A160893, J6(n)/J1(n) in  A160895, J7(n)/J1(n) in  A160897, J8(n)/J1(n) in  A160908, J9(n)/J1(n) in  A160953, J10(n)/J1(n) in  A160957, J11(n)/J1(n) in  A160960.


    Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in  A065958, J6(n)/J3(n) in  A065959, and J8(n)/J4(n) in  A065960.

    References

    Jordan's totient function Wikipedia
    (Text) CC BY-SA


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