On the local behavior of $\Psi (x,y)$

Abstract:

$\Psi (x,y)$ denotes the number of positive integers $\leqslant x$ and free of prime factors $> y$. In the range $y \geqslant \exp ({(\log \log x)^{5/3 + \varepsilon }})$, $\Psi (x,y)$ can be well approximated by a "smooth" function, but for $y \leqslant {(\log x)^{2 - \varepsilon }}$, this is no longer the case, since then the influence of irregularities in the distribution of primes becomes apparent. We show that $\Psi (x,y)$ behaves "locally" more regular by giving a sharp estimate for $\Psi (cx,y)/\Psi (x,y)$, valid in the range $x \geqslant y \geqslant 4\log x$, $1 \leqslant c \leqslant y$.