Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers

TL;DR

This paper establishes advanced distribution theorems for smooth numbers within a broader range than previously known, using zero-density and large sieve methods to handle their sparsity.

Contribution

It extends Bombieri--Vinogradov and Barban--Davenport--Halberstam theorems to larger ranges of y relative to x for smooth numbers, employing novel proof techniques.

Findings

Extended distribution theorems to larger y ranges for smooth numbers

Combined zero-density and large sieve methods effectively

Improved estimates for character sums over smooth numbers

Abstract

We prove Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for the y-smooth numbers less than x, on the range log^{K}x \leq y \leq x. This improves on the range \exp{log^{2/3 + \epsilon}x} \leq y \leq x that was previously available. Our proofs combine zero-density methods with direct applications of the large sieve, which seems to be an essential feature and allows us to cope with the sparseness of the smooth numbers. We also obtain improved individual (i.e. not averaged) estimates for character sums over smooth numbers.