Size of regulators and consecutive square-free numbers
Let \(\varepsilon_D=T+U\sqrt D>1\) be the fundamental solution to the Pell equation \(T^2-DU^2=1\) and let \[ S^f(x;\alpha):=\sharp \{D; \; D\leq x, D\neq \square, \varepsilon_D \leq D^{1/2+\alpha} \}. \] \textit{C. Hooley} [J. Reine Angew. Math. 353, 98--131 (1984; Zbl 0539.10019)] conjectured that there exist \(\delta(\alpha)\) and \(C(\alpha)\) such that \[ S^f(x;\alpha)\leq C(\alpha)x^{1-\delta(\alpha)} \] for every \(x\geq 2\). The main result of the paper under review is to show that for every \(\varepsilon >0\) \[ S^f(x;\alpha)\leq \mathcal O_\varepsilon\left(x^{\alpha/3+7/12+\varepsilon}\right) \] uniformly for \(\alpha\geq 0\) and \(x\geq 2\). The proof of their result relies on the square sieve introduced by \textit{D. R. Heath-Brown} [Math. Ann. 266, 251--259 (1984; Zbl 0514.10038)].
scientific article; zbMATH DE number 6149074