Multiplicative properties of the partition function (Q1110558)

Let \(p(n)\) denote the partition function and \(m(N,N+R)\) the number of multiplicatively independent values of p(n) with \(N\leq n\leq N+R\). The principal theorem of this paper is that there exists an \(N_ 0\) such that \[ m(N, N+R) \geq R\frac{\log N-\log R}{(3/2)\log N+R \log 2} \] for \(N\geq N_ 0\) and all natural numbers R, the proof of which involves a Hardy-Ramanujan formula for p(n). Several corollaries follow from this theorem, among them the following: Let a(n) be the number of nonisomorphic Abelian groups of order n, and let C(x) be the number of distinct values of a(n) for \(n\leq x\). Then for every \(\epsilon >0\) and \(x\geq x_ 1(\epsilon)\), \[ \log C(x) \geq (\log \log x)^ 2/(\log 16+\epsilon). \] If D(x) is the number of distinct a(n)\(\leq x\) with any n and \(x\geq x_ 2(\epsilon)\) then \[ \log D(x)\geq (\log \log x)^ 2/(\log 4+\epsilon). \]