On problem of S. Ramanujan. II.
type: Item
instance of: scholarly article
title: On problem of S. Ramanujan. II.
zbMATH Open document ID: 1216.11091
author: A. Smati
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2010-09-06T00:00:00Z
review text: In C. R., Math., Acad. Sci. Paris 340, No. 1, 1--4 (2005; Zbl 1062.11060), the author studied a problem posed by S. Ramanujan in 1915 consisting in finding the maximal order of the function \(d(d(n))\), where \(d(n)\) denotes the number of divisors of an integer \(n\) and improved the known bounds. In this note, the author studies the maximal order of \(d(d(n))\) as well as of the \(k\)-fold iteration of \(d(n)\). This generalization was formulated and studied by \textit{P. Erdős} and \textit{I. Kátai} [Fibonacci Q. 7, 267--274 (1969; Zbl 0188.34102)] and the author improved their result in (*) Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 29, 213--238 (2008; Zbl 1164.11059). In the paper under review using a different method, he proves the following theorem. 1) Let \(\varepsilon >0\) be an arbitrarily fixed real number. Then for \(n\) sufficiently large, one has \[ d(d(n)\leq e^{(1+\varepsilon)9\sqrt{192}\frac{\sqrt{\log n}}{\log\log n}}, \] and there exist infinitely many integers \(n\) such that \[ d(d(n)\geq e^{(1+\varepsilon)\log 2\frac{\sqrt{\log n}}{\log\log n}}. \] 2) For the \(k\)th iterate \((k\geq 3)\) the following estimates are obtained: \[ d_k(n)\leq e^{(1+\varepsilon)3a_k\frac{(\log n)^{1/F_k}}{\log\log n}}, \] and \[ d_k(n)\geq e^{(1+\varepsilon)b_k\log 2\frac{(\log n)^{1/F_k}}{\log\log n}}, \] where \(F_k\) denotes the \(k\)-th term of the Fibonacci sequence, and for \(k\) sufficiently large \(a_k\approx (254,947785\dots)(\frac{1+\sqrt 5}2)^k\), \(b_k\approx 0,495266\dots\) The constants \(a_k, b_k\) and their approximations have been given in (*).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11N64
zbMATH DE Number: 5779539
zbMATH Keywords: maximal order
zbMATH Keywords: iterated divisor function
MaRDI profile type: Publication
full work available at URL: j.crma.2010.07.018
OpenAlex ID: W1985228839
cites work: Q5621389
cites work: Q4735225
cites work: Q5582367
cites work: On a problem of S. Ramanujan
cites work: Q3539390
DOI: 10.1016/J.CRMA.2010.07.018
description: scientific article; zbMATH DE number 5779539