I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus linear forms plus a constant.

I would like to know what the current state of knowledge is for this sort of problem.

- This paper covers only the case of polynomials that "depend essentially on two variables". This has surely not been improved, else the H-L Conjecture E, F, etc. would have been solved.
- The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
- It does not give the constant (or prove that a constant exists!) for the densities it finds.

The third is my main interest at this point. References would be appreciated, whether to research papers, survey papers, or standard texts. I also have some interest in the cubic case, if anything is known. It seems that even characterizing which two-variable cubics represent an infinite number of primes is open...? Certainly the conditions Iwaniec gives for two-variable quadratics do not suffice to give an infinity of primes.

For example, with the polynomial $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$, setting $\Delta=b^2-4ac$ and $D=af^2-bef+ce^2+g\Delta$ and $C(x)=\sum_{p=P(x,y)\le x}$1, [1] shows that

$$\frac{x}{(\log x)^{1.5}}\ll C(x)\ll\frac{x}{(\log x)^{1.5}}$$

for all $P$ with $D\neq0$ and $\Delta$ not a square, but is it known that $C(x)\sim kx/(\log x)^{1.5}$ for some $k$? Are values of $e$ or $E$ known such that

$$e<\liminf\frac{C(x)(\log x)^{1.5}}{x}$$ or $$\limsup\frac{C(x)(\log x)^{1.5}}{x}< E$$ ?

Similarly, if $D=0$ or $\Delta$ is a square do we know when $$\ell=\lim\frac{C(x)\log x}{x}$$ exists and what its value is?

Related question: Who should I cite for these results? I'm never sure of when there might have been simultaneous discoveries or other reasons for priority issues.

[1] Iwaniec, H. (1974). Primes represented by quadratic polynomials in two variables. *Acta Arithmetica* **24**, pp. 435–459.

[2] Hardy, G. H., & Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. *Acta Mathematica* **44**:1, pp. 1-70.