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

• # Fulltext

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 $\alpha &gt; 0$ 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

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

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019