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Boletín de la Sociedad Matemática Mexicana

, Volume 24, Issue 2, pp 301–306 | Cite as

On the average value of the first n values of the Euler function

  • Ming-Liang Gong
  • Yong-Gao Chen
Original Article
  • 113 Downloads

Abstract

We prove that the number of positive integers \(n\le x\), such that \(\phi (1)+\phi (2)+\cdots +\phi (n)\) is a multiple of n, is less than \(x/(\log x)^{0.15742}\) for all sufficiently large x, where \(\phi \) stands for the Euler totient function.

Keywords

Euler totient function Prime factor 

Mathematics Subject Classification

11N37 

References

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    Balasubramanian, R., Luca, F., Ralaivaosaona, D.: Arithmetic properties of the sum of the first \(n\) values of the Euler function. Bol. Soc. Mat. Mex. 21, 9–17 (2015)MathSciNetCrossRefGoogle Scholar
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    Hardy, G.H., Ramanujan, S.: The normal number of prime factors of a number \(n\). Q. J. Math. 48, 76–92 (1917)zbMATHGoogle Scholar
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    Rhin, G., Viola, C.: On a permutation group related to \(\zeta (2)\). Acta Arith. 77, 23–56 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2017

Authors and Affiliations

  1. 1.School of Mathematical Sciences and Institute of MathematicsNanjing Normal UniversityNanjingPeople’s Republic of China

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