Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that
$$
\pi\_{2k}(x) \sim c\_{2k}\int\_2^x\frac{dt}{\log^2t}
$$
with
$$
c\_{2k}=2C\_2\prod\_{p|k,p>2}\frac{p-1}{p-2}
$$
where $\pi\_{2k}(x)$ is the count of such primes and $C\_2$ is the twin prime constant ([A005597](https://oeis.org/A005597)). i'm interested in the upper bound part.


I've seen a number of papers giving explicit constants $c$ such that $\pi\_2(x) \le c\int\_2^xdt/\log^2t$ for large enough $x$. What are the best known constants for $\pi\_{2k}$? I assume that they're no closer to the (conjectured) optimal constants than our best bounds for $\pi\_2$ (Wu [1] is within about 3.4).


I hope there has been some paper making this explicit? It would be fantastic to have a reference proving an upper bound of, say, $10c\_{2k}$, but I'm open to whatever can be found. (Perhaps some paper has an upper bound which is unbounded as $k\to\infty$ but finite for all $k$.)


[1] J Wu, [Chen's double sieve, Goldbach's conjecture, and the twin prime problem](https://arxiv.org/abs/0705.1652), *Acta Arithmetica* **114**:3 (2004), pp. 215–273. arXiv:0705.1652 [math.NT]



Best bound on $p, p+2k$ with $k$ fixed
J. R. Chen proved in his paper "On Goldbach's problem and the sieve methods" (Sci. Sinica Ser. A 21 (1978), 701-738.) that for $x>x\_0(k)$ we have
$$ \pi\_{2k}(x)<3.9171\times c\_{2k}\times\frac{x}{\log^2 x}.$$
I learned this from a paper of Pintz and Ruzsa (Acta Arith. 109 (2003), 169-194.).


