Asymptotics for smooth numbers in short intervals

TL;DR

This paper establishes asymptotic density results for smooth numbers in short intervals, extending known results and assuming the Riemann Hypothesis for broader ranges, using complex analysis and zero-density estimates.

Contribution

It provides new asymptotic formulas for smooth numbers in short intervals for a wider range of parameters, including under the Riemann Hypothesis.

Findings

Asymptotic density of smooth numbers matches long interval density for certain  in short intervals.

Results hold for all y  exp((log x)^{2/3+\u03b5}) without hypotheses.

Under RH, the results extend to smaller y, specifically y  ( log x)^K.

Abstract

A number is said to be $y$-smooth if all of its prime factors are less than or equal to $y.$ For all $17/30<\theta\leq 1,$ we show that the density of $y$-smooth numbers in the short interval $[x,x+x^{\theta}]$ is asymptotically equal to the density of $y$-smooth numbers in the long interval $[1,x],$ for all $y \geq \exp((\log x)^{2/3+\varepsilon}).$ Assuming the Riemann Hypothesis, we also prove that for all $1/2<\theta\leq 1$ there exists a large constant $K$ such that the expected asymptotic result holds for $y\geq (\log x)^{K}.$ Our approach is to count smooth numbers using a Perron integral, shift this to a particular contour left of the saddle point, and employ a zero-density estimate of the Riemann zeta function.