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
$\Psi (x,\,y)-\Lambda (x,\,y)$
.
Publisher
Cambridge University Press (CUP)