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

TL;DR

This paper refines the understanding of the relationship between smooth integers and de Bruijn's approximation, extending the range of validity and revealing connections to prime number theorem error terms and Chebyshev's bias.

Contribution

It extends the asymptotic equivalence range of $ ext{Psi}(x,y)$ and $ ext{Lambda}(x,y)$ by introducing a correction factor accounting for zeta zeros and prime powers.

Findings

Extended the range to $y \\ge (\\ ext{log} x)^{3/2+\\varepsilon}$

Uncovered a lower order term linked to prime number theorem error

Demonstrated Chebyshev's bias under Linear Independence hypothesis

Abstract

This paper is concerned with the relationship of $y$-smooth integers and de Bruijn's approximation $\Lambda(x,y)$. Under the Riemann hypothesis, Saias proved that the count of $y$-smooth integers up to $x$, $\Psi(x,y)$, is asymptotic to $\Lambda(x,y)$ when $y \ge (\log x)^{2+\varepsilon}$. We extend the range to $y \ge (\log x)^{3/2+\varepsilon}$ 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 $\Psi(x,y)/\Lambda(x,y)$. The term relates to the error term in the prime number 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 Independence hypothesis, we show a Chebyshev's bias in $\Psi(x,y)-\Lambda(x,y)$.