Frequency of gaps between consecutive prime numbers

Question

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.

Answer 1

Using the relationship $e^x=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{k=0}^K\frac{x^k}{k!}\right)$ leads to formula (1) below for $f_d(x)=\int\limits_2^x\frac{e^{-\frac{2 d}{\log (t)}}}{\log ^2(t)}\,dt$ which seems to converge fairly rapidly. The function $f_d(x)$ can also be approximated using the Mathematica $\text{NIntegrate}$ function as illustrated in formula (2) below.

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$$f_d(x)=\underset{K\to \infty }{\text{lim}}\left(\sum _{k=0}^K \frac{(2 d)^k (\Gamma (-k-1,-\log (x))-\Gamma (-k-1,-\log (2)))}{k!}\right)\tag{1}$$

$$f_d(x)\approx\text{NIntegrate}\left[\frac{e^{-\frac{2 d}{\log (t)}}}{\log ^2(t)},\{t,2,x\}\right]\tag{2}$$

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The following figure illustrates formula (1) for $f_1(x)$ above in orange overlaid on the blue reference function defined in formula (2) above where formula (1) is evaluated with the upper limit $K=100$.

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Figure (1): Illustration of formulas (1) and (2) for $f_1(x)$ in orange and blue respectively.

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The following figure illustrates formula (2) for $f_d(x)$ for $d\in \{1,2,3,4,5\}$ where the blue top curve corresponds to $f_1(x)$ and the purple bottom curve corresponds to $f_5(x)$.

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Figure (2): Illustration of formula (2) for $f_d(x)$ for $d\in \{1,2,3,4,5\}$

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Note $N(x,1)=\pi_2(x)$ where $\pi_2(x)$ counts the number of twin-prime pairs $(p, p+2)$ with $p\le x$ (see OEIS A071538). Also see Wikipedia: Twin prime and Weisstein, Eric W. "Twin Primes." From MathWorld--A Wolfram Web Resource.