A Problem of Ramanujan, Erdős, and Kátai on the Iterated Divisor Function

Abstract

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

The Maximal Order of Iterated Multiplicative Functions

“…In our approach, we assume that f (q ν ) = g(ν) for powers of primes q ∈ Q with a function g satisfying suitable axioms as listed below. By elaborating on the method of [4], we obtain upper and lower bounds on the maximal order of first iterates of arithmetic functions f which enjoy similar properties as those observed for d and δ, see Theorem 4.1 and Theorem 4.2.…”

“…Proof. (Compare [4,Lemma 3.3].) First note that the right-hand side of (6.2) is minimal if the ν j are decreasing.…”

“…log log N k holds. Erdős and Kátai [7], Ivić [14] and Smati [33,34] gave results on the maximal order, but a satisfying answer on the maximal order of the iterated divisor function was only given almost 100 years after Ramanujan's paper: Buttkewitz, Elsholtz, Ford and Schlage-Puchta [4] proved:…”

“…In [4], the case Q = P with the multiplicative arithmetic function d acting as d( p ν ) = ν + 1 is studied. In our approach, we assume that f (q ν ) = g(ν) for powers of primes q ∈ Q with a function g satisfying suitable axioms as listed below.…”