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On the values of the Euler function around shifted primes

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Abstract

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.

Résumé

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.

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Acknowledgements

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.

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Correspondence to Jean-Marie De Koninck.

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De Koninck, JM., Kátai, I. On the values of the Euler function around shifted primes. Ann. Math. Québec 43, 37–50 (2019). https://doi.org/10.1007/s40316-018-0101-x

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  • DOI: https://doi.org/10.1007/s40316-018-0101-x

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