I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM elliptic curves. Although I haven't yet worked out all the details but based on some recent papers by Wan/Xi and H. Qin it looks like it is quite possible to complete this work and establish such equivalence for all CM elliptic curves which means the difficult Lang-Trotter conjecture can be interpreted in terms of a relatively simple conjecture (only in the case of CM elliptic curves).


But I couldn't find much relevant work on this conjecture by Hardy and Littlewood. So could somebody refer to me any recent work on this conjecture or any promising work on this conjecture?



Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials
For the convenience of the reader, I have written Hardy and Littlewood's conjecture from their paper [linked in the comments above](https://link.springer.com/article/10.1007/BF02403921):


Suppose that $a,b,c$ are integers and $a$ is positive; that $\gcd(a,b,c) = 1$; that $a+b$ and $c$ are not both even; and that $D = b^2 - 4ac$ is not a square. Then there are infinitely many primes of the form $am^2 + bm + c$. The number $P(n)$ of such primes less than $n$ is given asymptotically by


$$\displaystyle P(n) \sim \frac{\varepsilon C}{\sqrt{a}} \frac{\sqrt{n}}{\log n} \prod\_{p | \gcd(a,b)} \left(\frac{p-1}{p} \right) $$


where $\varepsilon$ is $1$ if $a + b$ is odd and $2$ if $a+b$ is even. The constant $C$ is given by


$$\displaystyle C = \prod\_{w \geq 3, w \nmid a} \left(1 - \frac{1}{w-1} \left(\frac{D}{w} \right)\right).$$


To answer the question, there has not been any recent work on this problem for some time due to its perceived difficulty. In fact it is a long-standing debate between prominent analytic number theorists whether this conjecture is harder or easier than the twin prime conjecture. Despite the significant progress made by Zhang and Maynard about ten years ago on the latter, we are still very far away from twin primes themselves.


The best results towards this conjecture are due to Iwaniec ([Almost primes represented by quadratic polynomials](https://eudml.org/doc/142575)), who proved that the polynomial $x^2 + 1$ takes on infinitely many integer values with at most two prime divisors (including multiplicity) and that the density of such values is $\gg \sqrt{n}/\log(n)$. His arguments can be readily generalized to arbitrary quadratic polynomials satisfying the conditions imposed by Hardy and Littlewood above. On the other hand, Iwaniec's arguments have no chance of being improved to give actual primes without very significant new ideas, due to the parity barrier.


