Annales mathématiques du Québec

, Volume 43, Issue 1, pp 37–50 | Cite as

On the values of the Euler function around shifted primes

  • Jean-Marie De KoninckEmail author
  • Imre Kátai


Let \(\varphi \) stand for the Euler totient function. Garcia and Luca have proved that, given any positive integer \(\ell \), the set of those primes p such that \(\varphi (p+\ell )/\varphi (p-\ell )>1\) has the same density as the set of those primes p for which \(\varphi (p+\ell )/\varphi (p-\ell )<1\). Here we prove this result using classical results from probabilistic and analytic number theory. We then establish similar results for the sum of divisors function and for the k-fold iterate of the Euler function. We also examine the modulus of continuity of some arithmetical functions. Finally, we provide a general result regarding the existence of the distribution function for the function \(s(p):=f(p+\ell )-f(p-\ell )\) for any fixed positive integer \(\ell \) provided the additive function f satisfies certain conditions.


Euler totient function Distribution function Modulus of continuity 


Désignons par \(\varphi \) la fonction d’Euler. Garcia et Luca ont démontré que, étant donné un entier positif \(\ell \), l’ensemble des nombres premiers p tels que \(\varphi (p+\ell )/\varphi (p-\ell )>1\) admet la même densité que l’ensemble des nombres premiers p pour lesquels \(\varphi (p+\ell )/\varphi (p-\ell )<1\). Ici, nous démontrons ce résultat en utilisant certains résultats classiques de la théorie probabiliste et analytique des nombres. Nous établissons ensuite des résultats similaires pour la fonction somme de diviseurs et pour la k-ième itération de la fonction d’Euler. Nous examinons également le module de continuité de certaines fonctions arithmétiques. Enfin, nous établissons un résultat général concernant l’existence de la fonction de distribution de la fonction \(s(p):=f(p+\ell )-f(p-\ell )\) pour tout entier positif fixé \(\ell \), à condition que la fonction additive f satisfasse certaines propriétés.

Mathematics Subject Classification

11N60 11N25 26A15 



The authors would like to thank the referee for some helpful comments. The research of the first author was supported in part by a Grant from Canadian Network for Research and Innovation in Machining Technology, NSERC.


  1. 1.
    Deshouillers, J.M., Hassani, M.: A note on the distribution function of \(\varphi (p-1)/(p-1)\). J. Aust. Math. Soc. 93(1–2), 77–83 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Erdős, P.: On the distribution function of numbers of the form \(\sigma (n)/n\) and on some related questions. Pac. J. Math. 52, 59–65 (1974)CrossRefzbMATHGoogle Scholar
  3. 3.
    Erdős, P.: On the smoothness of the asymptotic distribution of additive arithmetical functions. Am. J. Math. 61, 722–725 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Erdős, P., Kátai, I.: On the concentration of distribution of additive functions. Acta Sci. Math. (Szeged) 41(3–4), 295–305 (1979)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Erdős, P., Wintner, A.: Additive arithmetical functions and statistical independance. Am. J. Math. 61, 713–721 (1939)CrossRefzbMATHGoogle Scholar
  6. 6.
    Garcia, S.R., Luca, F.: On the difference in values of the Euler totient function near prime arguments (2017). arxiv:1706.00392
  7. 7.
    Hildebrand, A.: Additive and multiplicative functions on shifted primes. Proc. Lond. Math. Soc. 59, 209–232 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Indlekofer, K.H., Kátai, I.: On the normal order of \(\varphi _{k+1}(n)/\varphi _k(n)\), where \(\varphi _k\) is the \(k\)-fold iterate of the Euler function. Liet. Mat. Rink. 44(1), 68–84 (2004) [translation in Lithuanian Math. J. 44(1), 47–61 (2004)]Google Scholar
  9. 9.
    Indlekofer, K.H., Kátai, I.: On the modulus of continuity of the distribution of some arithmetical functions. New Trends Probab Stat 223–231 (1992)Google Scholar
  10. 10.
    Kátai, I.: On distribution of arithmetical functions on the set prime plus one. Compos. Math. 19(4), 278–289 (1968)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lévy, P.: Sur les séries dont les termes sont des variables éventuelles indépendantes. Stud. Math. 3, 119–155 (1931)CrossRefzbMATHGoogle Scholar
  12. 12.
    Lukács, E.: Characteristic Functions. Griffin, London (1960)zbMATHGoogle Scholar
  13. 13.
    Tjan, M.M.: On the question of the distribution of values of the Euler function \(\varphi (n)\). Litovsk. Mat. Sb. 6, 105–119 (1966)MathSciNetGoogle Scholar

Copyright information

© Fondation Carl-Herz and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dép. de mathématiques et de statistiqueUniversité LavalQuébecCanada
  2. 2.Computer Algebra DepartmentEötvös Loránd UniversityBudapestHungary

Personalised recommendations