A problem of Ramanujan, Erdős, and Kátai on the iterated divisor function (Q2917607)

The authors determine the maximal order of \(d(d(n))\), where \(d(n)\) is the number of positive divisors of an integer \(n \,(\in\mathbb N)\). This is an old problem with a rich history, one of the difficulties being that \(d(d(n))\) is not multiplicative, whereas \(d(n)\) is. Already \textit{S. Ramanujan} in 1915 [Proc. Lond. Math. Soc. (2) 14, 347--409 (1915; JFM 45.1248.01), see also Collected papers of Srinivasa Ramanujan. Providence, RI: AMS Chelsea Publishing, 78--128 (2000; Zbl 1110.11001)] proved that, for infinitely many \(n\), NEWLINE\[NEWLINEd(d((n)) > 4^{\sqrt{2\log n}/\log\log n}.\leqno(1) NEWLINE\]NEWLINE \textit{P. Erdős} and \textit{I. Kátai} proved [Fibonacci Q. 7, 267--274 (1969; Zbl 0188.34102)] that NEWLINE\[NEWLINE d^{(k)}(n) < \exp\left((\log n)^{1/\ell_k+\varepsilon}\right) NEWLINE\]NEWLINE for fixed \(k\) and \(n\geq n_0(\varepsilon,k)\), and that for every \(\varepsilon>0\) NEWLINE\[NEWLINE d^{(k)}(n) > \exp\left((\log n)^{1/\ell_k-\varepsilon}\right) NEWLINE\]NEWLINE for infinitely many \(n\), where \(d^{(k)}(n)\) denotes the \(k\)-the iteration of \(d(n)\). Later \textit{P. Erdős} and the reviewer [Bull., Cl. Sci. Math. Nat., Sci. Math. 17, 13--22 (1989; Zbl 0695.10040)] improved the upper bound (when \(k=2\)) to NEWLINE\[NEWLINE \log d(d((n)) \ll \left(\frac{\log n\log\log n}{\log\log\log n}\right)^{1/2},\leqno(2) NEWLINE\]NEWLINE while \textit{A. Smati} [C. R., Math., Acad. Sci. Paris 340, No. 1, 1--4 (2005; Zbl 1062.11060) and Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 29, 213--238 (2008; Zbl 1164.11059)] obtained in (2) the upper bound \(\sqrt{\log n}\). It turned out that Ramanujan's bound in (1) is of the correct order of magnitude, as the present paper brings forth the result that NEWLINE\[NEWLINE \max_{n\leq x}\log d(d(n)) = \frac{\sqrt{\log x}}{\log\log x}\left(c + O\left(\frac{\log\log\log x}{\log\log x}\right)\right),\leqno(3) NEWLINE\]NEWLINE where NEWLINE\[NEWLINE c = \left(8\sum_{j=1}^\infty\log^2(1+1/j)\right)^{1/2} = 2.79598\ldots. NEWLINE\]NEWLINE This is in fact an asymptotic formula, which (up to the order of the error term in (3)) resolves the problem of the order of \(\max_{n\leq x}\log d(d(n))\). The authors also obtain another interesting result, involving a problem related to the previous one. Namely it is proved that (\(\omega(n)\) is the number of distinct prime factors of \(n\)) NEWLINE\[NEWLINE \max_{n\leq x}\omega(d(n)) =\frac{\sqrt{\log x}}{\log\log x}\left(\sqrt{8} + O\left(\frac{\log\log\log x}{\log\log x}\right)\right). NEWLINE\]NEWLINE The proof of (3), although elementary, is involved. First it is shown, by exhibiting an explicit sequence of \(n\) tending to infinity, that for this sequence the lower bound implied by (3) holds. The upper bound is more difficult and rests on four lemmas.