Number of twin primes

Consider number of twin primes less than $x$. We know that this number less than $\frac{Cx}{\log^2 x}$ for some constant $C$.

Denote by $p_n$ the $n$-th prime number. Do we have the same result about number of prime numbers $p_n<x$ such that $p_{n+1} - p_n < D$ for some constant $D$?

• Yes, in fact one can deduce such results for at least one even integer between 2 and 240 following the recent advances of James Maynard and the subsequent polymath project. – Stanley Yao Xiao Jun 13 '15 at 15:25
• See the "Added 2" section in my response for an effective upper bound due to J. R. Chen. – GH from MO Nov 3 '16 at 20:12

Yes, and this can be proved in the same way, e.g. by the large sieve or the Selberg sieve. Of course, the constant $C$ will depend on $D$.

For a lower bound (for $D=248$ at the moment), see my response to this question.

Added 1. As GuestPoster pointed out to me, the upper bound $Z(x)\ll x/\log^2 x$ for the number of twin primes up to $x$ was already established by Brun. His original paper is available here. In fact Brun shows $Z(x)<100x/\log^2 x$ for $x>x_0$, see the last display (37) in the paper. No doubt Brun's method works equally well for the original question above, with $100$ being replaced by an effective constant depending on $D$.

Added 2. J. R. Chen proved in his paper "On Goldbach's problem and the sieve methods" (Sci. Sinica Ser. A 21 (1978), 701-738.) that the number of solutions of $P-P'=h$ in primes $P'<P<x$ is less than $$7.8342\times \prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right)\prod_{\substack{p>2\\p\mid h}}\left(1+\frac{1}{p-2}\right)\times\frac{x}{\log^2 x},$$ assuming $x$ is sufficiently large in terms of $k$. I learned this from a paper of Pintz and Ruzsa (Acta Arith. 109 (2003), 169-194.).

• I am not sure what your "e.g." for "in the same way" means, as wasn't the twin primes upper bound of that form first proven by Brun in 1919/20? – GuestPoster Jun 14 '15 at 10:23
• @GuestPoster: Brun's upper bound was off by a factor of $(\log\log x)^2$ or so. At any rate, knowing that there are few twin primes does not imply aprior that there are few quasi-twin primes as well (with distance less than a thousand, say). Nevertheless, the techniques that provide the upper bound in the original post for the twin primes also provide the same kind of upper bound for the quasi-twin primes (with a different constant). I hope this clarifies. – GH from MO Jun 14 '15 at 10:57
• No, his first paper was off by that log-log power. A later paper even got the explicit constant $C=100$. Le crible d’Eratosthene et le theoreme de Goldbach. Comptes Rendus Acad. Sci. Paris, 168 (1919), 544-546 announces it, and Le crible d’Eratosthene et le theoreme de Goldbach. Videnselsk. Skr. 1, No. 3 (1920) gets 100. – GuestPoster Jun 14 '15 at 10:58
• @GuestPoster: I looked up Brun's paper, and you are right about his bound (not losing a loglog power). Let me update my post accordingly. – GH from MO Jun 14 '15 at 11:16
• I think Chen's result is only valid in the asymptotic regime where $x$ is assumed sufficiently large depending on $h$. – Terry Tao Nov 20 '16 at 18:19