A new elementary proof of the inequality φ(n) > π (n)

Carlo Sanna
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 3, Pages 35–37
Full paper (PDF, 131 Kb)

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Authors and affiliations

Carlo Sanna
Universitá degli Studi di Torino, Italy

Abstract

In this paper we provide a new elementary proof that the inequality φ(n) > π(n) holds for all integers n ≥ 91, an old result of L. Moser. Our proof is based on Bonse’s Inequality. This makes it somewhat simpler than Moser’s proof, which in turn relies on Bertrand’s Postulate.

Keywords

• Arithmetic functions
• Inequalities

AMS Classification

• 11A25

References

1. Archibald, C., Bertrand’s Postulate, Scripta Mathematica, Vol. 1, 1945, 109–120.
2. Bonse, H., Uer eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung, Arch. Math. Phys., Vol. 12, 1907, 292–295.
3. Moser, L., A theorem on the distribution of primes, Amer. Math. Monthly, Vol. 56, 1949, 624–625.
4. Moser, L., On the equation φ(n) = π (n) Pi Mu Epsilon J., 1951, 101–110.
5. Ramanujan, S., A proof of Bertrand’s Postulate, Journal of the Indian Mathematical Society, Vol. 11, 1919, 181–182.