I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand.
In (III), what does Zagier mean by "$\theta(x)$ changes by $O(\log x)$ if $x$ changes by $O(1)$?" I know the definition of Big O notation, but not what this grammatical construction means.
In (V), I am having trouble parsing the integral $$ \int_1^\infty \frac{d \theta(x)}{x^s}. $$ What does the notation $d \theta(x)$ mean? Is $\theta(x)$ the variable of integration in some sense? If so, how? (This may be a naive question.)
In the lower bound for $\theta(x)$ (on p. 707), in the bounds for the first summation -- $x^{1-\epsilon} \leq p \leq xx$ -- is $xx$ a typo, supposed to be $x$, or is it correct? If it's correct, why is the first inequality true?
I don't understand why the lower bound for $\theta(x)$, $$ \theta(x) \geq (1-\epsilon) \log x (\pi(x) + O(x^{1-\epsilon})), $$ is good enough. As $\epsilon \rightarrow 0$, $O(x^{1-\epsilon})$ tends to $O(x)$, and based on my interpretation of the previous summations, I expect this error term to be negative. Since $\pi(x) \leq x$, I basically don't understand why this is saying more than that $\theta(x) \geq 0$.
Thank you!