The Distribution of Totients

  • Kevin Ford
Part of the Developments in Mathematics book series (DEVM, volume 1)


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 V k (x), the number of mx 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 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 (math) and the bound lim inf x→∞ V 1(x)/V(x) ≤ 10- 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 pre-image 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.

Key words:

Euler’s function totients distributions Carmichaers Conjecture Sierpiński’s Conjecture 

1991 Mathematics Subject Classification:

Primary—11A25 11N64 


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  1. 1.
    R.C. Baker and G. Harman, “The difference between consecutive primes,” Proc. London Math. Soc. 72(3) (1996), 261–280.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    P.T. Bateman and R.A. Horn, “A heuristic asymptotic formula concerning the distribution of prime numbers,” Math. Comp. 16 (1962), 363–367.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    R.D. Carmichael, “On Euler’s ϕ-function,” Bull. Amer. Math. Soc. 13 (1907), 241–243.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    R.D. Carmichael, “Note on Euler’s ϕ-function,” Bull. Amer. Math. Soc. 28 (1922), 109–110.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    E. Cohen, “Arithmetical functions associated with the unitary divisors of an integer,” Math. Z. 74 (1960), 66–80.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    L.E. Dickson, “A new extension of Dirichlet’s theorem on prime numbers,” Messenger of Math. 33 (1904), 155–161.Google Scholar
  7. 7.
    P. Erdős, “On the normal number of prime factors of p — 1 and some related problems concerning Euler’s ϕ-function,” Quart. J. Math. (Oxford) (1935), 205–213.Google Scholar
  8. 8.
    P. Erdős, “Some remarks on Euler’s ϕ-function and some related problems,” Bull. Amer. Math. Soc. 51 (1945), 540–544.CrossRefMathSciNetGoogle Scholar
  9. 9.
    P. Erdős, “Some remarks on Euler’s ϕ-function,” Acta Arith. 4 (1958), 10–19.MathSciNetGoogle Scholar
  10. 10.
    P. Erdős and R.R. Hall, “On the values of Euler’s ϕ-function,” Acta Arith. 22 (1973), 201–206.MathSciNetGoogle Scholar
  11. 11.
    P. Erdős and R.R. Hall, “Distinct values of Euler’s ϕ-function,” Mathematika 23 (1976), 1–3.CrossRefMathSciNetGoogle Scholar
  12. 12.
    P. Erdős and C. Pomerance, “On the normal number of prime factors of ϕ(n)” ,Rocky Mountain J. of Math. 15 (1985), 343–352.CrossRefGoogle Scholar
  13. 13.
    K. Ford and S. Konyagin, “On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ,” Proceedings of the Number Theory Conference dedicated to Andrzej Schinzel on his 60th birthday (to appear).Google Scholar
  14. 14.
    K. Ford, “The number of solutions of ϕ(x) = mAnnals of Math. (to appear).Google Scholar
  15. 15.
    J. Friedlander, “Shifted primes without large prime factors,” Number theory and applications (Banff, AB, 1988), Kluwer Acad. Publ., Dorbrecht 1989, pp. 393–401.Google Scholar
  16. 16.
    H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.zbMATHGoogle Scholar
  17. 17.
    R.R. Hall and G. Tenenbaum, Divisors, Cambridge University Press, 1988.CrossRefzbMATHGoogle Scholar
  18. 18.
    G.H. Hardy and J.E. Littlewood, “Some problems of ‘Partitio Numerorum’: III. On the representation of a number as a sum of primes,” Acta Math. 44 (1923) 1–70.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    G.H. Hardy and S. Ramanujan, “The normal number of prime factors of a number n,” Quart. J. Math. 48 (1917), 76–92.zbMATHGoogle Scholar
  20. 20.
    A. Hildebrand and G. Tenenbaum, “Integers without large prime factors,” J. Théor. Nombres Bordeaux 5 (1993), 411–484.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    V. Klee, “On a conjecture of Carmichael,” Bull. Amer. Math. Soc. 53 (1947), 1183–1186.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    L.E. Mattics, “A half step towards Carmichael’s conjecture, solution to problem 6671,” Amer. Math. Monthly 100 (1993), 694–695.CrossRefMathSciNetGoogle Scholar
  23. 23.
    H. Maier and C. Pomerance, “On the number of distinct values of Euler’s ϕ-function,” Acta Arith. 49 (1988), 263–275.zbMATHMathSciNetGoogle Scholar
  24. 24.
    P. Masai and A. Valette, “A lower bound for a counterexample to Carmichael’s Conjecture,” Bollettino U.M.I. (1982), 313–316.Google Scholar
  25. 25.
    S. Pillai, “On some functions connected with ϕ(n),” Bull. Amer. Math. Soc. 35 (1929), 832–836.CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    C. Pomerance, “On the distribution of the values of Euler’s function,” Acta Arith. 47 (1986), 63–70.zbMATHMathSciNetGoogle Scholar
  27. 27.
    C. Pomerance, “Problem 6671,” Amer. Math. Monthly 98 (1991), 862.MathSciNetGoogle Scholar
  28. 28.
    A. Schinzel, “Sur l’equation ϕ(x) = m,” Elem. Math. 11 (1956), 75–78.zbMATHMathSciNetGoogle Scholar
  29. 29.
    A. Schinzel, “Remarks on the paper ‘Sur certaines hypothèses concernant les nombres premiers’“, Acta Arith. 7 (1961/62), 1–8.zbMATHMathSciNetGoogle Scholar
  30. 30.
    A. Schinzel and W. Sierpiński, “Sur certaines hypothèses concernant les nombres premiers,” Acta Arith. 4 (1958), 185–208.zbMATHMathSciNetGoogle Scholar
  31. 31.
    A. Schlafly and S. Wagon, “Carmichael’s conjecture on the Euler function is valid below 1010,000,000,” Math. Comp. 63 (1994), 415–419.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Kevin Ford
    • 1
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

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