Erdős Problem #1099

The problem

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

$$\liminf_{n\to \infty}h_\alpha(n) \ll_\alpha 1?$$

Statement quoted from erdosproblems.com/1099, maintained by Thomas Bloom. Original source: [Er81h,p.171]. Current status there: proved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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