In number theory, Jordan's totient function, denoted as J k ( n ) {\displaystyle J_{k}(n)} , where k {\displaystyle k} is a positive integer, is a function of a positive integer, n {\displaystyle n} , that equals the number of k {\displaystyle k} -tuples of positive integers that are less than or equal to n {\displaystyle n} and that together with n {\displaystyle n} form a coprime set of k + 1 {\displaystyle k+1} integers. Jordan's totient function is a generalization of Euler's totient function, which is the same as J 1 ( n ) {\displaystyle J_{1}(n)} . The function is named after Camille Jordan.
Definition For each positive integer k {\displaystyle k} , Jordan's totient function J k {\displaystyle J_{k}} is multiplicative and may be evaluated as
J k ( n ) = n k ∏ p | n ( 1 − 1 p k ) {\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right)\,} , where p {\displaystyle p} ranges through the prime divisors of n {\displaystyle n} .
Properties
∑ d | n J k ( d ) = n k . {\displaystyle \sum _{d|n}J_{k}(d)=n^{k}.\,}
which may be written in the language of Dirichlet convolutions as
J k ( n ) ⋆ 1 = n k {\displaystyle J_{k}(n)\star 1=n^{k}\,}
and via Möbius inversion as
J k ( n ) = μ ( n ) ⋆ n k {\displaystyle J_{k}(n)=\mu (n)\star n^{k}} . Since the Dirichlet generating function of μ {\displaystyle \mu } is 1 / ζ ( s ) {\displaystyle 1/\zeta (s)} and the Dirichlet generating function of n k {\displaystyle n^{k}} is ζ ( s − k ) {\displaystyle \zeta (s-k)} , the series for J k {\displaystyle J_{k}} becomes
∑ n ≥ 1 J k ( n ) n s = ζ ( s − k ) ζ ( s ) {\displaystyle \sum _{n\geq 1}{\frac {J_{k}(n)}{n^{s}}}={\frac {\zeta (s-k)}{\zeta (s)}}} . An average order of J k ( n ) {\displaystyle J_{k}(n)} is
J k ( n ) ∼ n k ζ ( k + 1 ) {\displaystyle J_{k}(n)\sim {\frac {n^{k}}{\zeta (k+1)}}} . The Dedekind psi function is
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