2024/09/09 by Younis, Khalid · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.05761
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<θ≤ 1, we show that the density of y-smooth numbers in the short interval [x,x+xθ] is asymptotically equal to the density of y-smooth numbers in the long interval [1,x], for all y ≥ exp((log x)2/3+ε). Assuming the Riemann Hypothesis, we also prove that for all 1/2<θ≤ 1 there exists a large constant K such that the expected asymptotic result holds for y≥ (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.