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Wikipedia / Jordan's totient function

In number theory, Jordan's totient function is a positive integer, is a function of a positive integer-tuples form a coprime set integers.

Jordan's totient function is a generalization of Euler's totient function. The function is named after Camille Jordan.

Definition

is multiplicative and may be evaluated as

  1. .

Properties

  1. which may be written in the language of Dirichlet convolutions as

  2. and via Möbius inversion as

    1. .

    Since the Dirichlet generating function becomes

    1. .

    An average order is

    1. .

    The Dedekind psi function is

    1. ,

    can also be shown to be integer-valued multiplicative functions.

  3. .

Order of matrix groups

The first two formulas were discovered by Jordan.

Examples

  • Explicit lists in the OEIS are J2 in , J3 in , J4 in , J5 in , J6 up to J10 in up to .

  • Multiplicative functions defined by ratios are J2(n)/J1(n) in , J3(n)/J1(n) in , J4(n)/J1(n) in , J5(n)/J1(n) in , J6(n)/J1(n) in , J7(n)/J1(n) in , J8(n)/J1(n) in , J9(n)/J1(n) in , J10(n)/J1(n) in , J11(n)/J1(n) in .

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

Notes

References