Jordan's totient function

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Let be a positive integer. In number theory, Jordan's totient function of a positive integer 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 a generalization of Euler's totient function, which is given by . The function is named after Camille Jordan.

Definition[edit]

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

, where ranges through the prime divisors of .

Properties[edit]

which may be written in the language of Dirichlet convolutions as[1]
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.
  • .      [2]

Order of matrix groups[edit]

  • The general linear group of matrices of order over has order[3]
  • The special linear group of matrices of order over has order
  • The symplectic group of matrices of order over has order

The first two formulas were discovered by Jordan.

Examples[edit]

Notes[edit]

  1. ^ Sándor & Crstici (2004) p.106
  2. ^ Holden et al in external links The formula is Gegenbauer's
  3. ^ All of these formulas are from Andrici and Priticari in #External links

References[edit]

  • L. E. Dickson (1971) [1919]. History of the Theory of Numbers, Vol. I. Chelsea Publishing. p. 147. ISBN 0-8284-0086-5. JFM 47.0100.04.
  • M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. Vol. 206. Springer-Verlag. p. 11. ISBN 0-387-95143-1. Zbl 0971.11001.
  • Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.

External links[edit]