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?

Source: erdosproblems.com/1099

Number theory
Divisors

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site credits Vose [Vo84]; the accepted claim is recorded on Vose's claim page (1984), and Tenenbaum's 1987 theorem, which settles the question along the factorials, the least common multiples and the primorials, on its own claim page (Tenenbaum, 1987).

Er81h: Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.
Te87: Tenenbaum, G., Sur un problème extrémal en arithmétique. Ann. Inst. Fourier (Grenoble) 37 (1987), no. 2, 1--18; doi:10.5802/aif.1083.
Vo84: Vose, Michael D., Integers with consecutive divisors in small ratio. J. Number Theory (1984), 233-238.

Problem 1099

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