This plot, from Odlyzko, Rubinstein & Wolf, 1999, shows the frequency of the gaps between consecutive primes of a given size.
Where $N(x,d)$ is the number of primes $p \leq x$ such that $p+2d$ is the smallest prime $> p$.
My question is what happens to the plot as $x \to \infty$? In particular, does the gradient of the plot approach a specific value? What about $\frac{N(x,d)}{\pi(x)}$ ?
This explanation is provided in the paper:
Figure 1 Log plot of $\log {N(x,d)}$, the number of jumps of length $2d$ between primes up to $x$, for $x = 2^{20}, 2^{22}, \ldots, 2^{44}$. Integrating (2-8) by parts and taking logarithms we see that, for fixed $x$, the graph should follow a straight line with small perturbations of size $\log{A_{d,1}}$ (see text); this can be seen in the figure.
Where (2-8) is the approximation: $$N(d,x) \sim A_{d,1} \int_2^x \frac{\exp{(-2d / \log t )}}{\log^2{t}} dt .$$
I am having difficulty performing the integration
\begin{align} \int \frac{\exp{(-2d / \log t )}}{\log^2{t}} dt &= \int \frac{t}{2d} \frac{2d\exp{(-2d / \log t )}}{t\log^2{t}} dt \\ &= \int \frac{t}{2d} \frac{d}{dt} \left[ \exp{(-2d / \log t )} \right] dt \\ &= \frac{t}{2d} \exp{(-2d / \log t )} - \int \frac{1}{2d} \left[ \exp{(-2d / \log t )} \right] dt \end{align}
and I don't see how to make progress.