Smooth numbers in arithmetic progressions to large moduli
Number Theory
2025-09-17 v3
Abstract
We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size . This overcomes a longstanding barrier of present in previous works of Bombieri-Friedlander-Iwaniec, Fouvry-Tenenbaum, Drappeau, and Maynard. We build on Drappeau's variation of the dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers-Iwaniec type estimates for sums of Kloosterman sums.
Keywords
Cite
@article{arxiv.2304.11696,
title = {Smooth numbers in arithmetic progressions to large moduli},
author = {Alexandru Pascadi},
journal= {arXiv preprint arXiv:2304.11696},
year = {2025}
}
Comments
51 pages. v2: Improved main result using an adjusted arrangement of exponential sums. v3: Incorporates referees' comments (to appear in Compos. Math.)