About the Hardy-Littlewood conjecture by [Terence Tao](https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-461113):


**Conjecture 2** (Prime tuples conjecture, quantitative form) Let ${k\_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed admissible ${k\_0}$-tuple. Then the number of natural numbers ${n < x}$ such that ${n+{\mathcal H}}$ consists entirely of primes is ${({\mathfrak G} + o(1)) \frac{x}{\log^{k\_0} x}}$.


It was written there: From the methods of sieve theory, one can obtain an upper bound of ${(C\_{k\_0} {\mathfrak G} + o(1)) \frac{x}{\log^{k\_0} x}}$ for the number of ${n < x}$ with ${n + {\mathcal H}}$ all prime, where ${C\_{k\_0}}$ depends only on ${k\_0}$. Sieve theory can also give analogues of Conjecture 2 if the primes are replaced by a suitable notion of almost prime (or more precisely, by a weight function concentrated on almost primes).


**Questions:**


1) Where can I find these results? Could you give a reference?


2) How does ${C\_{k\_0}}$ depend on ${k\_0}$?


Thank you!



Upper bound for the first Hardy-Littlewood conjecture
This is explained for example in Iwaniec & Kowalski's "Analytic Number Theory", as an standard application of Selberg's $\Lambda^2$ sieve. See chapter 6, Elementary sieve methods. In particular you get $C\_{k\_0}=2^{k\_0}k\_0!$, for an upper bound, using your notation:


$${(2^{k\_0}k\_0! {\mathfrak G} + o(1)) \frac{x}{\log^{k\_0} x}}$$


Of course, the conjecture predicts that you can take $C\_{k\_0}=1$, but as far as I know the $2^{k\_0}k\_0!$ bound still is the best one known unconditionally.


Probably the most complete classic reference is


* Atle Selberg, Lectures on Sieves (1991)


You can also prove the same bound using Montgomery and Vaughan's large sieve. For this and all other things sieve-related, Opera de Cribro is he best source.


