In number theory, Jordan's totient function, denoted as , where is a positive integer, is a function of a positive integer, , that equals the number of -tuples of positive integers that are less than or equal to and that together with form a coprime set of integers.
Jordan's totient function is a generalization of Euler's totient function, which is the same as . The function is named after Camille Jordan.
Definition
For each positive integer , Jordan's totient function is multiplicative and may be evaluated as
, where ranges through the prime divisors of .
Properties
which may be written in the language of Dirichlet convolutions as
:
and via Möbius inversion as
:.
Since the Dirichlet generating function of is and the Dirichlet generating function of is , the series for becomes
:.
:.
:,
and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of ), the arithmetic functions defined by or can also be shown to be integer-valued multiplicative functions.
Order of matrix groups
The first two formulas were discovered by Jordan.
Examples
- Explicit lists in the OEIS are J<sub>2</sub> in , J<sub>3</sub> in , J<sub>4</sub> in , J<sub>5</sub> in , J<sub>6</sub> up to J<sub>10</sub> in up to .
- Multiplicative functions defined by ratios are J<sub>2</sub>(n)/J<sub>1</sub>(n) in , J<sub>3</sub>(n)/J<sub>1</sub>(n) in , J<sub>4</sub>(n)/J<sub>1</sub>(n) in , J<sub>5</sub>(n)/J<sub>1</sub>(n) in , J<sub>6</sub>(n)/J<sub>1</sub>(n) in , J<sub>7</sub>(n)/J<sub>1</sub>(n) in , J<sub>8</sub>(n)/J<sub>1</sub>(n) in , J<sub>9</sub>(n)/J<sub>1</sub>(n) in , J<sub>10</sub>(n)/J<sub>1</sub>(n) in , J<sub>11</sub>(n)/J<sub>1</sub>(n) in .
- Examples of the ratios J<sub>2k</sub>(n)/J<sub>k</sub>(n) are J<sub>4</sub>(n)/J<sub>2</sub>(n) in , J<sub>6</sub>(n)/J<sub>3</sub>(n) in , and J<sub>8</sub>(n)/J<sub>4</sub>(n) in .
Notes
References
External links