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]

  • 2
    $\begingroup$ 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

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.).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.