totient

A totient is a sequence $f:\{1,2,3,\mathrm{\dots }\}\to ℂ$ suchthat

$$g\ast f=h$$

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 isseldom used nowadays except in two cases. The Euler totient $\varphi$satisfies

$${\iota }_0\ast \varphi ={\iota }_1$$

where ${\iota }_k$ denotes the function $n\mapsto n^k$ (which is completelymultiplicative). The more general Jordan totient $J_k$ is defined by

$${\iota }_0\ast J_k={\iota }_k.$$