Smooth integers and de Bruijn's approximation λ

This paper is concerned with the relationship of -smooth integers and de Bruijn's approximation. Under the Riemann hypothesis, Saias proved that the count of -smooth integers up to, is asymptotic to when. We extend the range to by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers. We use this correction term to uncover a lower order term in the asymptotics of. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of lead to large positive (resp. negative) values of, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in.