On some conjectures of Erdős in Astérisque. I
Let \(A(d,\alpha)\) denote the density of the integers having a divisor \(D \equiv 1\pmod d\) in the range \(1 < D < \exp(d^ \alpha)\). \textit{P. Erdős} [Astérisque 61, 73-82 (1979; Zbl 0399.10001)] conjectured the existence of \(\alpha_ 0 > 1\) with the property that \(A(d,\alpha) \to 0\) or 1 as \(d \to \infty\), depending on whether the fixed value \(\alpha\) satisfies \(\alpha < \alpha_ 0\) or \(\alpha > \alpha_ 0\). The author not only proves the conjecture, with \(\alpha_ 0 = 1/\log 2\), but also makes some progress with the deeper problem of allowing \(\alpha\) to depend on \(d\), and letting \(\alpha \to \alpha_ 0\) as \(d \to \infty\). More specifically, his first theorem states that if \(\alpha < 1/\log 2 - \sqrt{c\log\log\log d/\log d}\), where \(c > 2/\log 2\), then \(A(d,\alpha) \to 0\) as \(d \to \infty\). Next, let \(A^*(d,\alpha)\) be the density of the integers \(n\) such that, for every \(\ell\) prime to \(d\), the number \(n\) has a divisor \(D \equiv \ell \pmod d\) in the interval \(\exp\exp(\sqrt{\log d}) < D < \exp(d^ \alpha)\). The second theorem states that if \(\alpha = 1/\log 2+\xi(d)/\sqrt{\log d}\), where \(\xi(d) \to \infty\), then \(A^*(d,\alpha) \to 1\) as \(d \to \infty\). The proofs depend on theorems in probability theory, namely the law of the iterated logarithms, and a theorem of Erdős and Rényi in probabilistic group theory, respectively.
scientific article; zbMATH DE number 98394
scientific article