The distribution of smooth numbers in arithmetic progressions

TL;DR

This paper proves that smooth numbers are uniformly distributed in arithmetic progressions for a broad range of moduli, with implications for number theory conjectures, and identifies subgroup structures for larger moduli.

Contribution

It extends the range of moduli for which smooth numbers are equidistributed and introduces subgroup structures for larger moduli, improving previous results.

Findings

Smooth numbers are equidistributed mod q for q< y^{4√e - ε}

Existence of large subgroups where smooth numbers are equidistributed within cosets

Implications for Vinogradov's conjecture on quadratic non-residues

Abstract

For a wide range of $x$ and $y$ we show that ${\Cal S}(x,y)$, the set of integers below $x$ composed only of prime factors below $y$, is equidistributed in the reduced residue classes $\pmod q$ for all $q<y^{4\sqrt{e}-\epsilon}$. This improves earlier work of Granville; any improvement of this range of $q$ would have interesting consequences for Vinogradov's conjecture on the least quadratic non-residue. For larger ranges of $q$ we prove the existence of a large subgroup of the group of reduced residues such that ${\Cal S}(x,y)$ is equidistributed within cosets of that subgroup.