If we define 
$$\pi^2(N)=\vert [p: p\leq N, p\in\mathbb{P}, p-2\in\mathbb{P}]\vert$$
where $\mathbb{P}$ is the set of all primes (as the number of twin primes less than $N$), and we define
$$\pi^D(N)=\vert [p: p\leq N, p\in\mathbb{P}, Dp-2\in\mathbb{P}]\vert$$


Can anyone think of how to show that for $D$ prime and large enough with respect to $N$ and $N$ large enough, we have $\pi^2(N)\geq \pi^D(N)$. It seems heuristically true, simply because the probability, given a prime $p$, that $Dp-2$ is also prime is less than the probability that $p-2$ (which is smaller than $Dp-2$ and thus within a denser area of primes) is also prime.  

 Sieve methods allow us to conclude that the upper bounds of both sets are multiples of each other, but not much else. Any ideas?



Twin primes and D primes
What you are calling $\pi^2(N)$ should really be $\pi^1(N).$ It is slightly unfortunate that this is the number of twin primes up to $N$ which is commonly written as $\pi\_2(N),$ but that is just notation. There is no reason to think that $D$ being prime will make a difference. It will be clearer to let $D$ be *any* nonegative integer and


$$\pi^D(N)=\begin{cases}
\vert [p: p\leq N, p\in\mathbb{P}, Dp-2\in\mathbb{P}]\vert& \text{if $D$ is odd},\\\
\vert [p: p\leq N, p\in\mathbb{P}, Dp-1\in\mathbb{P}]\vert& \text{if $D$ is even}.
\end{cases}$$


For "large" $N$ relative to $D$ there will be hardly any difference between the density of primes near $N$ and near $DN:$ $$\frac1{\ln(N)}-\frac1{\ln(DN)}<\frac{\ln(D)}{(\ln{N})^2}.$$ 
What should really matter is the set of odd primes dividing $D$ (the prime $2$ is built in.) 


Recall that the number of primes up to $N$ is $\pi(N)\sim \frac{N}{\ln{N}}.$ It is quite true that no one (yet) can prove that $\lim\_{N \to \infty}\pi^1(N)=\infty$, but it is known how fast it goes there. Less humorously: it is widely believed that the number of twin prime pairs up to $N$ is $\pi^1(N)=\pi\_2(N)=C\_2\frac{N}{(\ln{N})^2}$ for the constant $C\_2 = \prod\_{p\ge 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016.$ 


We can discuss the present question with a high confidence that computational results will support us. In the spirit of the amazing paper [Heuristic Reasoning in the Theory of Numbers](http://www.computing-wisdom.com/jstor/heuristic-number-theory.pdf) (read it!) one might define $R\_D(N)=\frac{\pi^D(N)}{\pi^1(N)}$ and experiment to make conjectures. I will predict that for large enough $N$ $R\_2,R\_4,R\_8 \sim 1$ , $R\_3,R\_6,R\_9,R\_{12},R\_{18} \sim 2$ 
and in general $R\_D$ will be very close to the $R\_d$ from Polya's paper. This will take very large $N$ for larger $D.$ The same phenomenon should be easier to observe for $\pi^D(M,N)$ with the condition on $p$ changed to $M \le p \le N.$


Here is a very small experiment. The first $2000$ primes following $10,000,000$ run from $M=10000019$ to $N=10032181.$ The formula number of twin primes predicts $\pi\_2(N)-\pi\_2(M)=143.27.$
Here is a sorted list of $[D,\pi^D(M,N)]$ for selected values.


[3, 329], [6, 305], [9, 296], [24, 282], [12, 280], [18, 276], [27, 259], [36, 242], [5, 218], [10, 188], [25, 182], [20, 176], [1, 169], [2, 166], [40, 163], [16, 152], [4, 140], [8, 137], [32, 135]


The values are grouped as predicted. The values for $1,2,4,8,16,32$ were supposed to all be around $143.27.$ The evidence given is fair but maybe not overwhelming. Here are the ratios $\frac{\pi^D}{143.27}$


[3, 2.295882763], [6, 2.128401954], [9, 2.065596650], [24, 1.967899512], [12, 1.953942777], [18, 1.926029309], [27, 1.807397069], [36, 1.688764829], [5, 1.521284020], [10, 1.311933008], [25, 1.270062805], [20, 1.228192603], [1, 1.179344033], [2, 1.158408932], [40, 1.137473831], [16, 1.060711793], [4, .9769713887], [8, .9560362875], [32, .9420795534]


I leave more extensive calculations to the motivated reader.


