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[1]
- and via Möbius inversion as
- .
- Since the Dirichlet generating function of is and the Dirichlet generating function of is , the series for becomes
- .
- An average order of is
- .
- The Dedekind psi function is
- ,
- 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
- 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
- Explicit lists in the OEIS are J2 in OEIS: A007434, J3 in OEIS: A059376, J4 in OEIS: A059377, J5 in OEIS: A059378, J6 up to J10 in OEIS: A069091 up to OEIS: A069095.
- Multiplicative functions defined by ratios are J2(n)/J1(n) in OEIS: A001615, J3(n)/J1(n) in OEIS: A160889, J4(n)/J1(n) in OEIS: A160891, J5(n)/J1(n) in OEIS: A160893, J6(n)/J1(n) in OEIS: A160895, J7(n)/J1(n) in OEIS: A160897, J8(n)/J1(n) in OEIS: A160908, J9(n)/J1(n) in OEIS: A160953, J10(n)/J1(n) in OEIS: A160957, J11(n)/J1(n) in OEIS: A160960.
- Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in OEIS: A065958, J6(n)/J3(n) in OEIS: A065959, and J8(n)/J4(n) in OEIS: A065960.
Notes
- ^ Sándor & Crstici (2004) p.106
- ^ Holden et al in external links. The formula is Gegenbauer's.
- ^ All of these formulas are from Andrica and Piticari in #External links.
References
- 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
- Andrica, Dorin; Piticari, Mihai (2004). "On some extensions of Jordan's arithmetic functions". Acta Universitatis Apulensis. 7: 13–22. MR 2157944.
- Holden, Matthew; Orrison, Michael; Vrable, Michael. "Yet Another Generalization of Euler's Totient Function" (PDF). Archived from the original (PDF) on 2016-03-05. Retrieved 2011-12-21.