PlanetMath: totient

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totient

(Definition)

A totient is a sequence $f:{\{1,2,3,\ldots\}}\to {\mathbb{C}}$ such that

$\displaystyle g\ast f=h$

and

, where $\ast$ denotes the convolution product (or Dirichlet product; see multiplicative function).

The term `totient' was introduced by Sylvester in the 1880's, but is seldom used nowadays except in two cases. The Euler totient $\phi$ satisfies

$\displaystyle \iota_0\ast\phi = \iota_1$

where $\iota_k$ denotes the function $n\mapsto n^k$ (which is completely multiplicative). The more general Jordan totient

is defined by

$\displaystyle \iota_0\ast J_k=\iota_k.$

"totient" is owned by mathcam. [ owner history (1) ]

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Also defines:

totient, Jordan totient

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Cross-references: function, Euler, multiplicative function, product, completely multiplicative, sequence
There are 8 references to this entry.

This is version 2 of totient, born on 2003-05-25, modified 2003-05-25.
Object id is 4293, canonical name is Totient.
Accessed 4351 times total.

Classification:

11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

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