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

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    • 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

  • Proceedings – Mathematical Sciences | News

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      Posted on July 25, 2019

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