The maximal size of the k-fold divisor function for very large k

Let d(k)(n) denote the number of ways of writing n as an (ordered) product of k positive integers. When k = 2, Wigert proved in 1907 that log d(k)(n) <= (1 +o(1)) log k log n/log log n (n -> infinity). (*) In 1992, Norton showed that (*) holds whenever k = o(log n); this is sharp, since (*) holds with equality when n is a product of the first several primes. In this note, we determine the maximal size of log d(k)(n) when k >> log n. To illustrate: Let kappa > 0 be fixed, and let k, n -> infinity in such a way that k/ log n -> kappa; then log d(k)(n) <= ( s + kappa Sigma(p prime) Sigma(l >= 1)1/lpls + o(1) ) log n, where s > 1 is implicitly defined by Sigma(p prime) log p/p(s) -1 = 1/kappa. Moreover, this upper bound is optimal for every value of kappa. Our results correct and improve on recent work of Fedorov.