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)

Read more about this topic:  Jordan's Totient Function

Famous quotes containing the word properties:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)