The sequence \(x/n\) and its subsequences
A positive integer \(n\) is said to be sparsely totient if \(\varphi(n)< \varphi(m)\) for all integers \(m>n\), where \(\varphi\) is Euler's function. One of the interesting problems concerning sparsely totient numbers \(n\) is to find the smallest number \(\lambda>0\) such that \[ \max_{p\mid n}p= O_\varepsilon ((\log n)^{\lambda+\varepsilon}), \] for all such \(n\), where \(p\) denotes a prime and \(\varepsilon>0\) is arbitrary. In this paper the authors prove that \(\lambda<37/20\). Since both the above problem and a problem on the Thue equation are linked to the behaviour of the sequences of the form \(x/r\) for given real \(x\), and natural numbers \(r\) in a suitable interval, in this paper, the authors also investigate a more general problem on these sequences. They prove the following result by a method which is adapted from \textit{W. Schmidt}'s beautiful work [in Small fractional parts of polynomials, Regional Conf. Ser. Math. 32, Am. Math. Soc. (1977; Zbl 0362.10032)]. Let \(R\) be a natural number, \({\mathbf x}\in\mathbb{R}^t\) and \(|{\mathbf x}|\leq R^{c_1}\). Let \(Q\) be a natural number with \(1<c_2\leq Q\leq R^{c_3}\) and let \(\psi_1,\dots, \psi_t\) be positive numbers satisfying \(\psi_j\leq c_4(\log Q)^{-t}\) for \(j=1,\dots,t\), and \(\psi_1\dots\psi_t= Q^{-1}\). Then \[ |x_j/r|\leq \psi_j, \qquad j=1,\dots, t \] for some integer \(r\) satisfying \(R\leq r\leq 2QR\), where \(|\alpha|\) denotes the distance from the real \(\alpha\) to the nearest integer. Here \(c_j\) are positive effectively computable constants, \(c_3\) depends on \(c_1\) and \(c_2\) and \(c_4\) depend on \(c_1\) and \(t\).
scientific article; zbMATH DE number 1026389
scientific article