On the maximal order of numbers in the "factorisatio numerorum" problem

Abstract

Let m(n) be the number of ordered factorizations of n in factors larger than 1. We prove that for every eps>0 n^{rho} m(n) < exp[(log n)^{1/rho}/(loglog n)^{1+eps}] holds for all integers n>n_0, while, for a constant c>0, n^{rho} m(n) > exp[c(log n)^{1/ρ}/(loglog n)^{1/rho}] holds for infinitely many positive integers n, where rho=1.72864... is the real solution to zeta(rho)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).

We have rewritten the paper and improved considerably the lower bound. Thus now we know that the max. order of m(n) is n^{rho}/(exp((log n)^{1/rho+o(1)})). Submitted to JNT