In addition to the Euler totient function, there are a great many generalizations and related functions which go by the "totient", usually with some name: Jordan, Lehmer\*, Schemmel, Nagell, Alder, Lucas, Stevens, Eugeni–Rizzi, Holden–Orrison–Vrable, Cohen, Menon, Garcia–Ligh, von Sterneck, etc.


Is there some unifying reason that these are all called totient totient functions? I'm familiar with the (mainly historical) use of "totient" to mean "numbers less than or equal to and coprime to" but it's not obvious how that connects to the generalized functions. (Possibly there is no deeper connection than "related to Euler's function".)


\* [Derrick Norman Lehmer](http://en.wikipedia.org/wiki/Derrick_Norman_Lehmer), not to be confused with his son [Derrick Henry Lehmer](http://en.wikipedia.org/wiki/Derrick_Henry_Lehmer).



What is a totient?
*What is a totient?* Here is one [answer](http://www.hindawi.com/journals/ijmms/1996/129618/abs/) that might have the generality you seek:



> 
> An arithmetical function is said to be a totient if it is the
>  Dirichlet convolution between a completely multiplicative function and
>  the inverse of a completely multiplicative function.
> 
> 
> 


For the origin of the name (Sylvester's nomenclature for Euler's counting function of relative primality), see this [StackExchange Q&A,](http://english.stackexchange.com/questions/23694/where-does-the-word-totient-come-from) with a warning not to interpret the name too literally: *as with most mathematical coinages, there is little that is meaningful in Sylvester's metaphor that corresponds to 'relative primality'*


