The largest prime factor of the integers in an interval (Q1352509)

The main result in this paper is that the largest prime factor of integers in an interval of the form \([x,x+x^{1/2+\varepsilon}]\), with \(x\) sufficiently large, is at least \(x^{11/12-\varepsilon}\). The previously known best lower bound was \(x^{0.82}\) due to \textit{A. Balog}, \textit{G. Harman} and \textit{J. Pintz} [J. Lond. Math. Soc., II. Ser. 28, 218-226 (1983; Zbl 0514.10034)]. These authors used sieve methods, whereas the present method is more analytic, related to that of \textit{M. Jutila} [J. Indian Math. Soc., New Ser. 37 (1973), 43-53 (1974; Zbl 0317.10049)]. A further advance in this problem was made recently by \textit{G. Harman} [Sieve methods, exponential sums and their applications in number theory, Cambridge Univ. Press, 161-173 (1997)], who obtained the lower bound \(x^{19/20}\) by sieve methods.