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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). 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, Acta Arithmetica 114:3 (2004), pp. 215–273. arXiv:0705.1652 [math.NT]

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    $\begingroup$ The Article by Bateman and Horn from 1961 gives an upper bound right on the first page with some references, which in this case is 8 times the expected constant. Don't know how much state of the art this is. jstor.org/stable/2004056?seq=1#page_scan_tab_contents $\endgroup$ – Jarek Kuben Nov 20 '16 at 5:08
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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.).

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