Bounding Zaremba’s conjecture

We prove in particular that for any sufficiently large prime $p$ there is $1\leq a \leq p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. 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.