On problem of S. Ramanujan. II.
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 (*).
scientific article; zbMATH DE number 5779539