On a number theoretic inequality of Ramanujan
type: Item
instance of: scholarly article
title: On a number theoretic inequality of Ramanujan
author: Horst Alzer
author: Mehdi Hassani
published in: Applied Mathematics E-Notes
publication date: 2024-10-08T00:00:00Z
review text: In this short but interesting article, the authors prove that, for all integers \(n \geqslant 2\), we have \N\[\N\tau(n) \leqslant \left( 1+\frac{\log n}{\log \gamma(n)}\right)^{\omega(n)}\N\]\Nprovided that, if \(n = p_1^{\alpha_1} \dotsb p_r^{\alpha_r}\), then \(p_1 < \dotsb < p_r\) and \(\alpha_1 \leqslant \dotsb \leqslant\alpha_r\). Here, \(\tau(n)\), \(\gamma(n)\) and \(\omega(n)\) are respectively the number of divisors, the squarefree kernel and the number of distinct prime divisors of \(n\). This improves on an old result of Ramanujan [\textit{S. Ramanujan}, Proc. Lond. Math. Soc. (2) 14, 347--409 (1915; JFM 45.1248.01)]. The proof rests on Chebyshev's inequality dealing with the product of arithmetic means, which can be found for instance in [\textit{D. S. Mitrinović}, Analytic inequalities. In cooperation with P. M. Vasić. Berlin: Springer-Verlag (1970; Zbl 0199.38101)], Theorem 1 p. 36.
reviewed by: Olivier Bordellès
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 7923067
zbMATH Keywords: inequalities for arithmetic functions
zbMATH Keywords: Chebyshev's inequality
MaRDI profile type: Publication
cites work: Q4276250
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cites work: Analytic Inequalities
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description: scientific article; zbMATH DE number 7923067