On smooth square-free numbers in arithmetic progressions

Booker and Pomerance [Proc. Amer. Math. Soc. 145 (2017) 5035–5042] have shown that any residue class modulo a prime (Formula presented.) can be represented by a positive (Formula presented.) -smooth square-free integer (Formula presented.) with all prime factors up to (Formula presented.) and conjectured that in fact one can find such (Formula presented.) with (Formula presented.). Using bounds on double Kloosterman sums due to Garaev [Mat. Zametki 88 (2010) 365–373] we prove this conjecture in a stronger form (Formula presented.) and also consider more general versions of this question replacing (Formula presented.) -smoothness of (Formula presented.) by the stronger condition of (Formula presented.) -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 (Formula presented.) which is (Formula presented.) -smooth. Additionally, we obtain stronger results for almost all primes (Formula presented.).