Open Cover Balls in Analytic Theorem I am having trouble to understand the definition of cover, sub-cover and compactness which I find in the definition of analytic theorem (used for prime number theorem). Can anyone help me understand those three definition according to diagram used in analytic theorem given below? 

To be precise, where are the line segment $L$ in the picture, and where are those open cover balls?
 A: The second sentence defines $L$ to be the set of complex numbers with magnitude no larger than $R$ which have zero real part (so they lie on the y-axis). In other words, it's the line segment between $-Ri$ and $Ri$.
Oddly, the drawing shows this as the horizontal axis, because the lines $t=\pm R$ are shown as vertical lines (note $t= \operatorname{Im} s$, so one would expect these to be horizontal lines with constant imaginary part). The balls covering $L$ are shown in the drawing as the disks along the horizontal axis between $t=-R$ and $t=R$ .
So either the drawing is wrong (probable), or the description is wrong (possible), or the drawing has an unorthodox orientation of the axes (doubtful).

The take-away: In the drawing, $L$ is supposed to be the line segment along the x-axis, and the small balls shown are the finite subcover of $L$ chosen from some arbitrary open cover. $L$ is the horizontal diameter of the circle of radius $R$. As I mentioned, the drawing and the description don't match up.
