Some multiplicative properties of products of natural numbers in a short interval
Let \(\varepsilon > 0\) be a small real number and \(x > x_0 (\varepsilon)\). The author proves that the interval \((x,x + zx^\varepsilon]\) contains many \(z\)-smooth integers, that is integers, all of whose prime factors are at most \(z\), where \(z \geq x^{11/26 + \varepsilon}\). The result is based on a deep estimate of \textit{H. Iwaniec} and \textit{C. J. Mozzochi} [J. Number Theory 29, 60-93 (1988; Zbl 0644.10031)]. The above result has various consequences of the following type. Let \(k(n)\) denote the smallest positive integer \(k\) such that the real numbers \(\log (n + 1), \dots, \log (n + k)\) are linearly dependent over the rational numbers. We have \(k(n) \leq n^{11/26 + \varepsilon}\).
prime divisors of integers in short intervals
scientific article; zbMATH DE number 848930