The distribution of totients

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This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler's ϕ-function. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of ϕ(x) =m (the “multiplicity” of m), and Vk(x), the number of m ≤ x with A(m) = k. The first of the main results of the paper is a determination of the true order of V (x). It is also shown that for each k ≥ 1, if there is a totient with multiplicity k, then Vk(x) V (x). We further show that every multiplicity k ≥ 2 is possible, settling an old conjecture of Sierpiński. An older conjecture of Carmichael states that no totient has multiplicity 1. This remains an open problem, but some progress can be reported. In particular, the results stated above imply that if there is one counterexample, then a positive proportion of all totients are counterexamples. Determining the order of V (x) and Vk(x) also provides a description of the “normal” multiplicative structure of totients. This takes the form of bounds on the sizes of the prime factors of a pre-image of a typical totient. One corollary is that the normal number of prime factors of a totient ≤ x is c log log x, where c ≈ 2:186. Lastly, similar results are proved for the set of values taken by a general multiplicative arithmetic function, such as the sum of divisors function, whose behavior is similar to that of Euler's function.

Original languageEnglish (US)
Pages (from-to)27-34
Number of pages8
JournalElectronic Research Announcements of the American Mathematical Society
Issue number5
StatePublished - Apr 27 1998
Externally publishedYes


  • Carmichael's conjecture
  • Distributions
  • Euler's function
  • Sierpi_nski's conjecture
  • Totients

ASJC Scopus subject areas

  • Mathematics(all)


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