So I have come across the proof of Prime Number Theorem by Newman. I do understand the most part but except for the part in which Newman proves Tauberian's theorem. I have two current questions for the theorem and the proof. The first question is that what does it mean by extend holomorphically for $\displaystyle G(z) = \int_{0}^{\infty}F(t) e^{-zt} \mathrm{d}t$ where $F(t)$ is bounded and locally integrable ? Cause from what I have learned in $\textbf{Complex Analysis}$, the meaning would be equivalent to analytic continuation, wouldn't it?
The second question is referring to the proof of the theorem, so Newman considered the contour which is the boundary of the region $D_R = \{z \in \mathbb{C} | \Re{(z)} \geq - \delta, \vert z \vert \leq R \}$ in which $\delta > 0$ is an arbitrary small number. My inquiry is that how can he consider such $\delta$ exists for any functions under the above restrictions? Cause if one takes $F(t) = 1$ then $G(z)$ will not converge, won't it? I'm a self-learner, if there are any misconceptions, hope you all can point them out. Thank you very much.