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

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

2005-05-17

arxiv.org/abs/math/0505352v2

Mathematics

Number Theory

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}

Scientific paper

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/\rho}/(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).

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