Jordan's Totient Function - Properties

Properties

which may be written in the language of Dirichlet convolutions as

and via Möbius inversion as

.

Since the Dirichlet generating function of μ is 1/ζ(s) and the Dirichlet generating function of nk is ζ(s-k), the series for Jk becomes

.
  • An average order of Jk(n) 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 p-k), the arithmetic functions defined by or can also be shown to be integer-valued multiplicative functions.

  • \sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)

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