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

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