We have to change the proof of Lemma 3 only slightly.
Unfortunately, the notation from number theory slightly conflicts with the notation from probability theory.
Slightly refining a result of , Davenport proved ......
The estimates (5) then follow by slightly increasing the value of $m$.
They are all slightly different.
Note that (2) is a slightly weakened version of the Pólya inequality.
The motivation for the results of this section is the following result of John (paraphrased slightly to suit our purposes).