Erdős Problem #1099

Let \(1=d_1<\cdots<d_{\tau(n)}=n\) be the divisors of \(n\), and for \(\alpha>1\) let \[h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.\] Is it true that \[\liminf_{n\to \infty}h_\alpha(n) \ll_\alpha 1?\] Erdős [Er81h] remarks that \(n!\) or the least common multiple of \(\{1,\ldots,n\}\) would be good candidates for an infinite sequence of \(n\) with \(h_\alpha(n)\) bounded. The \(\liminf\) is trivially \(\geq 1\), just considering the term \(i=1\). A positive answer to the main question was provided by Vose [Vo84] by constructing a specific sequence. Tenenbaum [Te87] proved that the two explicit sequences mentioned above satisfy this property. The statement "\(\liminf h_\alpha(n)\) is finite" is formalised as "\(h_\alpha(n)\leq C\) for infinitely many \(n\)".