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* = {p ∈ P; 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 j ≠ k, j, k = 1, 2, …, r. In this paper we prove that if n ∈ Z_{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 m ∈ Z_{s}^{+}\left( P^{\ast }\right).

Keywords

  • Arithmetical functions
  • Mąkowski–Schinzel conjecture

AMS Classification

  • 11A25
  • 11N37

References

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Cite this paper

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.

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