On smooth square-free numbers in arithmetic progressions

TL;DR

This paper proves that for large primes, every residue class can be represented by a square-free number with controlled size and smoothness, improving previous bounds and extending to more general smoothness conditions.

Contribution

It establishes new bounds on the size and smoothness of square-free numbers representing residue classes modulo primes, confirming and strengthening earlier conjectures.

Findings

Representation of all residue classes by square-free numbers with size up to p^{2+o(1)}

Improved bounds on smoothness for representing residue classes

Stronger results hold for almost all primes p

Abstract

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^{\alpha}$-smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer $s\le p^{2+o(1)}$ which is $p^{1/(4e^{ /2})+o(1)}$-smooth. Additionally, we obtain stronger results for almost all primes $p$.