Integers without large prime factors in short intervals: Conditional results

Abstract

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by \(\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}\) , we show that for any given α > 0 the interval \((X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]\) contains an integer having no prime factor exceeding Xα for all X sufficiently large.