Sur un problème extrémal en arithmétique. (On an extremal problem in arithmetic)
Let \(F_{\alpha}(n)=\sum (d_{i+1}/d_ i-1)^{\alpha}\), where \(d_ i\) runs through the sequence of divisors of n in their natural order. Erdős had conjectured that, for any fixed \(\alpha >1\), (i) \(F_{\alpha}(n)\) is bounded on an infinite sequence of values n, and (ii) \(F_{\alpha}(n)\) is bounded uniformly for all values n of the form \(n=k!\), \(n=[1,2,...,k]\) and \(n=p_ 1...p_ k,\) where \(p_ i\) denotes the i-th prime. (i) was recently proved by \textit{M. D. Vose} [J. Number Theory 19, 233-238 (1984; Zbl 0543.10031)]. In the present paper the author proves Erdős' conjecture in its strong version (ii), and in fact a generalization of it. The proof is different from Vose's. The author first reduces the problem to that of estimating the number of divisors of n in short intervals near \(\sqrt{n}\). He then applies Fourier techniques to obtain the required estimates for integers n that are of the above form.
scientific article; zbMATH DE number 4008498
scientific article