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Sep 19, 2018 at 17:12 comment added hao chen Thank you for the answer! It is kind of surprising to me that simple Taylor approximation is not too far away from this bound. The error is $K^n/n!$, which is roughly $(eK/n)^n / \sqrt{2\pi n}$. Well, still there is a $2^n$ factor difference.
Sep 19, 2018 at 17:07 vote accept hao chen
Sep 16, 2018 at 19:36 comment added user111 @Nate Eldredge I don't have the paper by Batyrev so I can't say, but several subsequent papers cite his paper for the mentioned result. I have added a reference to one of these papers in the answer. In particular, it has a proof of Batyrev's result.
Sep 16, 2018 at 19:30 history edited user111 CC BY-SA 4.0
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Sep 16, 2018 at 15:42 comment added Nate Eldredge Are you sure that this paper contains a complete proof? Sometimes short papers in Doklady were just research announcements. Also, since older Russian journals can be hard for many people to obtain, do you know if the proof is written down somewhere else (preferably in English)?
Sep 16, 2018 at 15:11 history answered user111 CC BY-SA 4.0