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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, not to be confused with his son Derrick Henry Lehmer.

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    $\begingroup$ A linguistic remark: The Latin word "totiens" (or "toties") means "that/so many times". Similarly "quotie(n)s?" means "how many times?" and most numerals have similar endings: "decie(n)s" means "ten times". $\endgroup$
    – Joonas Ilmavirta
    Commented Dec 22, 2014 at 19:27
  • $\begingroup$ @Joonas: so "totient function" means a function counting the number of times something happens? $\endgroup$
    – Qiaochu Yuan
    Commented Dec 22, 2014 at 21:49
  • $\begingroup$ @QiaochuYuan, so it seems. There is also a Latin word "tot" (so many) instead of "totiens" (so many times); I don't know if this choice was conscious. There are four kinds of numerals in Latin: for example the words "unus--primus--singuli--semel" translate to "one--first--one at a time--once". That aside, I don't see how to explain connections to Latin so that both "quotient" and "totient" make sense. They don't really mean a question and an answer in mathematics, do they? I assume those words were introduced to mathematical English separately, disregarding the original semantic relation. $\endgroup$
    – Joonas Ilmavirta
    Commented Dec 22, 2014 at 22:07
  • $\begingroup$ @JoonasIlmavirta -- at least quotient makes sense, as the answer to how many times the denominator fits into the numerator? (perhaps you meant something still different). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Dec 23, 2014 at 1:56
  • $\begingroup$ @WłodzimierzHolsztyński, both quotient and totient make sense as something describing how many times something happens. But from the linguistic point of view I see no obstruction to swapping the mathemaical meanings of these words; both words seem to refer to the number of times something happens (which is understandable), neither one being more of a question or an answer than the other. The transfer from Latin words to English mathematical terms does not seem to preserve the semantic relations of the two words (it's not functorial?) although it preserves rough meaning (it's continuous?). $\endgroup$
    – Joonas Ilmavirta
    Commented Dec 24, 2014 at 22:15

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What is a totient? Here is one answer 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, 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'

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    $\begingroup$ I think the question is why such a general definition deserves the name "totient." Is it just by analogy with the Euler totient function or is there some general non-mathematical meaning of the word "totient" that is being invoked here? $\endgroup$
    – Qiaochu Yuan
    Commented Dec 22, 2014 at 21:48
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    $\begingroup$ The paper answers my question perfectly! (Thanks also for the nomenclature note.) $\endgroup$
    – Charles
    Commented Dec 23, 2014 at 0:32

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