Smooth numbers and the Dickman ρ function

Abstract

We establish an asymptotic formula for ψ(x, y) whose shape is xρ(log x/ log y) times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for ψ(x, y). With this formula at hand we prove oscillation results for ψ(x, y), which resolve a question of Hildebrand on the range of validity of ψ(x, y) ≍ xρ(log x/ log y). We also address a question of Pomerance on the range of validity of ψ(x, y) ≥ xρ(log x/ log y).

Along the way we improve classical estimates for ψ(x, y) and, on the Riemann Hypothesis, uncover an unexpected phase transition of ψ(x, y)at y = (log x)^3/2+o(1).