The Distribution of Totients
 Kevin Ford
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Abstract
This paper is a comprehensive study of the set of totients, i.e., the set of values taken by Euler’s Φfunction. 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 V _{ k }(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 V _{ k } (x) ≫ V(x). Sierpiński conjectured that every multiplicity k ≥ 2 is possible, and we deduce this from the Prime ktuples 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 (math) and the bound lim inf_{ x→∞} V _{1}(x)/V(x) ≤ 10^{5.000.000.000} is shown. Determining the order of V(x) and V _{ k }(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 preimage of a typical totient. One corollary is that the normal number of prime factors of a totient ≤ x is c log logx, 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.
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 Title
 The Distribution of Totients
 Book Title
 Analytic and Elementary Number Theory
 Book Subtitle
 A Tribute to Mathematical Legend Paul Erdös
 Pages
 pp 67151
 Copyright
 1998
 DOI
 10.1007/9781475745078_8
 Print ISBN
 9781441950581
 Online ISBN
 9781475745078
 Series Title
 Developments in Mathematics
 Series Volume
 1
 Series ISSN
 13892177
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag US
 Additional Links
 Topics
 Keywords

 Euler’s function
 totients
 distributions
 Carmichaers Conjecture
 Sierpiński’s Conjecture
 Primary—11A25
 11N64
 Industry Sectors
 eBook Packages
 Editors
 Authors

 Kevin Ford ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA
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