Mathematische Annalen

, Volume 374, Issue 1–2, pp 253–271 | Cite as

Popular subsets for Euler’s \(\varphi \)-function

  • Paul PollackEmail author


Let \(\varphi (n) = \#(\mathbb {Z}/n\mathbb {Z})^{\times }\) (Euler’s totient function). Let \(\epsilon > 0\), and let \(\alpha \in (0,1)\). We prove that for all \(x > x_0(\epsilon ,\alpha )\) and every subset \(\mathscr {S}\) of [1, x] with \(\#\mathscr {S}\le x^{1-\alpha }\), the number of \(n\le x\) with \(\varphi (n)\in \mathscr {S}\) is at most \(x/L(x)^{\alpha -\epsilon }\), where
$$\begin{aligned} L(x) = \exp (\log x\cdot \log _3{x}/\log _2 x). \end{aligned}$$
Under plausible conjectures on the distribution of smooth shifted primes, this upper bound is best possible, in the sense that the number \(\alpha \) appearing in the exponent of L(x) cannot be replaced by anything larger.

Mathematics Subject Classification

Primary 11N64 Secondary 11N25 11N37 



The author is supported by NSF award DMS-1402268. He thanks the referee for a careful reading of the manuscript and helpful comments.


  1. 1.
    Baker, R.C., Harman, G.: Shifted primes without large prime factors. Acta Arith. 83, 331–361 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Banks, W.D., Ford, K., Luca, F., Pappalardi, F., Shparlinski, I.E.: Values of the Euler function in various sequences. Monatsh. Math. 146, 1–19 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banks, W.D., Friedlander, J.B., Pomerance, C., Shparlinski, I.E.: Counting integers with a smooth totient, submitted arXiv:1809.01214 [math.NT]
  4. 4.
    Banks, W.D., Friedlander, J.B., Pomerance, C., Shparlinski, I.E.: Multiplicative structure of values of the Euler function, High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, Amer. Math. Soc., Providence, RI, pp. 29–47 (2004)Google Scholar
  5. 5.
    Berend, D., Tassa, T.: Improved bounds on Bell numbers and on moments of sums of random variables. Probab. Math. Statist. 30, 185–205 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Canfield, E.R., Erdős, P., Pomerance, C.: On a problem of Oppenheim concerning “factorisatio numerorum”. J. Number Theory 17, 1–28 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carmichael, R.D.: Note on Euler’s \(\varphi \)-function. Bull. Amer. Math. Soc. 28, 109–110 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erdős, P.: On the normal number of prime factors of \(p-1\) and some related problems concerning Euler’s \(\varphi \)-function. Quart. J. Math. 6, 205–213 (1935)CrossRefzbMATHGoogle Scholar
  9. 9.
    Erdős, P.: On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen 4, 201–206 (1956)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Erdős, P.: Solution of two problems of Jankowska. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6, 545–547 (1958)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Erdős, P.: Some remarks on Euler’s \(\varphi \) function. Acta Arith. 4, 10–19 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Erdős, P., Pomerance, C.: On the normal number of prime factors of \(\varphi (n)\). Rocky Mountain J. Math. 15, 343–352 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ford, K.: The distribution of totients. Ramanujan J. 2, 67–151 (1998). see arXiv:1104.3264 [math.NT] for updates and corrections
  14. 14.
    Ford, K.: The number of solutions of \(\varphi (x)=m\). Ann. of Math. 150(2), 283–311 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hall, R.R., Tenenbaum, G.: Divisors, Cambridge Tracts in Mathematics, vol. 90. Cambridge University Press, Cambridge (1988)Google Scholar
  16. 16.
    Hardy, G.H., Ramanujan, S.: The normal number of prime factors of a number \(n\). Quart. J. Math. 48, 76–92 (1917)zbMATHGoogle Scholar
  17. 17.
    Knopp, M.I.: Modular functions in analytic number theory. Markham, Chicago (1970)zbMATHGoogle Scholar
  18. 18.
    Luca, F., Pollack, P.: An arithmetic function arising from Carmichael’s conjecture. J. Théor. Nombres Bordeaux 23, 697–714 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Luca, F., Pomerance, C.: On the average number of divisors of the Euler function. Publ. Math. Debrecen 70, 125–148 (2007). errata in 89 (2016), 257–260MathSciNetzbMATHGoogle Scholar
  20. 20.
    Oppenheim, A.: On an arithmetic function. J. London Math. Soc. 1, 205–211 (1926)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Oppenheim, A.: On an arithmetic function (II). J. London Math. Soc. 2, 123–130 (1927)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pollack, P.: How often is Euler’s totient a perfect power? J. Number Theory 197, 1–12 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pollack, P., Vandehey, J.: Some normal numbers generated by arithmetic functions. Canad. Math. Bull. 58, 160–173 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pomerance, C.: Popular values of Euler’s function. Mathematika 27, 84–89 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pomerance, C.: On the distribution of pseudoprimes. Math. Comp. 37, 587–593 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pomerance, C., Two methods in elementary analytic number theory, Number theory and applications (Banff, AB, : NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ. Dordrecht 1989, 135–161 (1988)Google Scholar
  27. 27.
    Tenenbaum, G.: Introduction to analytic and probabilistic number theory, third ed., Graduate Studies in Mathematics, vol. 163, American Mathematical Society, Providence, RI (2015)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA

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