On the composition of the functions σ and φ on the set Zs+(P*)

Aleksander Grytczuk
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 1, Pages 14—18
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Authors and affiliations

Aleksander Grytczuk
Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
4a Prof. Szafrana Str., 65-516 Zielona Góra, Poland

Abstract

In 1964, A. Mąkowski and A. Schinzel ([8], Cf.[6]) conjectured that for all positive integers m, we have
\frac{\sigma \left( \varphi \left( m\right) \right) }{m}\geq \frac{1}{2}, (*)
where σ denote the sum of divisors function and φ is the Euler’s totient function.
Let P be the set of all odd primes and
P* = {pP; p = 2αk + 1; α ≥ 1; k > 1; (k,2) = 1}.
Moreover, let
Z_{s}^{+}\left( P^{\ast }\right) =\left\{n=\dprod \limits_{j=1}^{r}p_{j};p_{j}=2^{\alpha _{j}}m_{j}+1;\alpha _{j}\geq 1,m_{j}>1,p_{j}\in P^{\ast }\right\}
where (mj, mk) = 1; for all jk, j, k = 1, 2, …, r. In this paper we prove that if nZ_{s}^{+}\left( P^{\ast }\right)
then we have \frac{\sigma \left( \varphi \left( n\right) \right) }{n}\geq 1. From this and Sándor’s result it follows that (*) is true for all positive integers m ≥ 1 such that the squarefree part of mZ_{s}^{+}\left( P^{\ast }\right).

Keywords

  • Arithmetical functions
  • Mąkowski—Schinzel conjecture

AMS Classification

  • 11A25
  • 11N37

References

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  10. Sándor, J. On the composition of some arithmetic functions, Studia Babes-Bolyai University, Mathematica, Vol. 34, 1989, 7–14.
  11. Sándor, J. Remarks on the functions σ(n) and φ(n), Babes-Bolyai Univ. Seminar on Math. Analysis, Vol. 7, 1989, 7–12.
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Cite this paper

APA

Grytczuk, A. (2013). On the composition of the functions σ and φ on the set Zs+(P*), Notes on Number Theory and Discrete Mathematics, 19(1), 14-18.

Chicago

Grytczuk, Aleksander. “On the Composition of the Functions σ and φ on the Set Zs+(P*).” Notes on Number Theory and Discrete Mathematics 19, no. 1 (2013): 14-18.

MLA

Grytczuk, Aleksander. “On the Composition of the Functions σ and φ on the Set Zs+(P*).” Notes on Number Theory and Discrete Mathematics 19.1 (2013): 14-18. Print.

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