On smooth square-free numbers in arithmetic progressions

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 $p$.

Using the ideas indicated at the end of Version 1, we have now obtained a series of improvements of our preliminary results

References in corpus (3)

• Sumsets of reciprocals in prime fields and multilinear Kloosterman sums
• A multiplicative analogue of Schnirelmann's theorem
• A Note on the least squarefree number in an arithmetic progression

Cited by in corpus (1)

• Subproducts of small residue classes