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

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 $y\ge (\log x{)}^{2+\epsilon }$. We extend the range to $y\ge (\log x{)}^{3/2+\epsilon }$ 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 $\Psi (x,y)-\Lambda (x,y)$.

arXiv comment22 pages. Accepted version

References in corpus (1)

• Uniform estimates for smooth polynomials over finite fields