On Korobov bound concerning Zaremba's conjecture

We prove in particular that for any sufficiently large prime $p$ there is $1\le a<p$ such that all partial quotients of $a\mathrm{/}p$ are bounded by $O(\log p\mathrm{/}\log \log p)$. For composite denominators a similar result is obtained. This improves the well–known Korobov bound concerning Zaremba’s conjecture from the theory of continued fractions.