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Hometotient

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# totient

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

$g\ast f=h$ |

for some two completely multiplicative sequences $g$ and $h$, 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

$\iota_{0}\ast\phi=\iota_{1}$ |

where $\iota_{k}$ denotes the function $n\mapsto n^{k}$ (which is completely
multiplicative). The more general *Jordan totient* $J_{k}$ is defined by

$\iota_{0}\ast J_{k}=\iota_{k}.$ |

Defines:

totient, Jordan totient

Type of Math Object:

Definition

Major Section:

Reference

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