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    • Integers without Large Prime Factors in Short Intervals: Conditional Results

      Goutam Pal Satadal Ganguly

      Volume 120 Issue 5 November 2010 pp 515-524

    • Fulltext

       

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/120/05/0515-0524

    • Keywords

       

      Smooth numbers; Riemann zeta function.

    • Abstract

       

      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.

    • Author Affiliations

       

      Goutam Pal1 Satadal Ganguly2

      1. 5/1, Nandaram Sen, First Lane, Kolkata 700 005, India
      2. School of Mathematics, Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Mumbai 400 005, India

    • Dates

       
      Published
      November 2010
      Manuscript received
      15 June 2010
      Manuscript revised
      13 July 2010

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