The distribution of totients

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This paper is a comprehensive study of 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 ≤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). Sierpiński conjectured that every multiplicity k ≥ 2 is possible, and we deduce this from the Prime k-tuples Conjecture. 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. The lower bound for a possible counterexample is extended to 101010 and the bound lim infx→∞ V1(x)/V(x) ≤ 10-5,000,000,000 is shown. 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)67-151
Number of pages85
JournalRamanujan Journal
Issue number1-2
StatePublished - 1998
Externally publishedYes


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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