Integers without Large Prime Factors in Short Intervals: Conditional Results
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Under the Riemann hypothesis and the conjecture that the order of growth of the argument of $\zeta(1/2+it)$ is bounded by $(\log t)^{\frac{1}{2}+o(1)}$, we show that for any given $\alpha > 0$ the interval $(X, X+\sqrt{X}(\log X)^{1/2+o(1)}]$ contains an integer having no prime factor exceeding $X^\alpha$ for all 𝑋 sufficiently large.
Volume 131, 2021
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