Jordan's totient function

For each positive integer ${\displaystyle k}$ , Jordan's totient function ${\displaystyle J_{k}}$  is multiplicative and may be evaluated as

${\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right)\,}$ , where ${\displaystyle p}$  ranges through the prime divisors of ${\displaystyle n}$ .

• ${\displaystyle \sum _{d|n}J_{k}(d)=n^{k}.\,}$ 

which may be written in the language of Dirichlet convolutions as^[1]

${\displaystyle J_{k}(n)\star 1=n^{k}\,}$ 

and via Möbius inversion as

${\displaystyle J_{k}(n)=\mu (n)\star n^{k}}$ .

Since the Dirichlet generating function of ${\displaystyle \mu }$  is ${\displaystyle 1/\zeta (s)}$  and the Dirichlet generating function of ${\displaystyle n^{k}}$  is ${\displaystyle \zeta (s-k)}$ , the series for ${\displaystyle J_{k}}$  becomes

${\displaystyle \sum _{n\geq 1}{\frac {J_{k}(n)}{n^{s}}}={\frac {\zeta (s-k)}{\zeta (s)}}}$ .

• An average order of ${\displaystyle J_{k}(n)}$  is

${\displaystyle J_{k}(n)\sim {\frac {n^{k}}{\zeta (k+1)}}}$ .

• The Dedekind psi function is

${\displaystyle \psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}}$ ,

and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of ${\displaystyle p^{-k}}$ ), the arithmetic functions defined by ${\displaystyle {\frac {J_{k}(n)}{J_{1}(n)}}}$  or ${\displaystyle {\frac {J_{2k}(n)}{J_{k}(n)}}}$  can also be shown to be integer-valued multiplicative functions.

• ${\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}$ .^[2]

• The general linear group of matrices of order ${\displaystyle m}$  over ${\displaystyle \mathbf {Z} /n}$  has order^[3]

${\displaystyle |\operatorname {GL} (m,\mathbf {Z} /n)|=n^{\frac {m(m-1)}{2}}\prod _{k=1}^{m}J_{k}(n).}$ 

• The special linear group of matrices of order ${\displaystyle m}$  over ${\displaystyle \mathbf {Z} /n}$  has order

${\displaystyle |\operatorname {SL} (m,\mathbf {Z} /n)|=n^{\frac {m(m-1)}{2}}\prod _{k=2}^{m}J_{k}(n).}$ 

• The symplectic group of matrices of order ${\displaystyle m}$  over ${\displaystyle \mathbf {Z} /n}$  has order

${\displaystyle |\operatorname {Sp} (2m,\mathbf {Z} /n)|=n^{m^{2}}\prod _{k=1}^{m}J_{2k}(n).}$ 

The first two formulas were discovered by Jordan.

• 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 J_2k(n)/J_k(n) are J₄(n)/J₂(n) in OEIS: A065958, J₆(n)/J₃(n) in OEIS: A065959, and J₈(n)/J₄(n) in OEIS: A065960.

• 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.

• 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.