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This paper is concerned with the relationship of $y$-smooth integers and deBruijn's approximation $\Lambda(x,y)$. Under the Riemann hypothesis, Saiasproved that the count of $y$-smooth integers up to $x$, $\Psi(x,y)$, isasymptotic to $\Lambda(x,y)$ when $y \ge (\log x)^{2+\varepsilon}$. We extendthe range to $y \ge (\log x)^{3/2+\varepsilon}$ by introducing a correctionfactor that takes into account the contributions of zeta zeros and primepowers.  We use this correction term to uncover a lower order term in the asymptoticsof $\Psi(x,y)/\Lambda(x,y)$. The term relates to the error term in the primenumber theorem, and implies that large positive (resp. negative) values of$\sum_{n \le y} \Lambda(n)-y$ lead to large positive (resp. negative) values of$\Psi(x,y)-\Lambda(x,y)$, and vice versa. Under the Linear Independencehypothesis, we show a Chebyshev's bias in $\Psi(x,y)-\Lambda(x,y)$.

Smooth integers and de Bruijn's approximation $\Lambda$