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; DOI: 10.4064/aa-24-5-435-459.
[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.
Best Answer
The modern reference work on the subject seems to be [1], but it spends only a page and a half on the subject of primes in multivariate quadratic polynomials (pp. 396-397). More than half this space is devoted to Iwaniec's 1974 result. The balance mentions Sarnak's application to the Problem of Apollonius and a result of "J. Cho and H. Kim" on counting primes in $\mathbb{Q}[\sqrt{-2}].$ So nothing there.
Pleasants [2] shows that, subject to a Davenport-Lewis [3] condition on the $h^*$ (a complexity measure on the cubic form part), multivariate cubic polynomials have the expected number of primes. Unfortunately this condition requires (as a necessary but insufficient condition) that there be at least 8 variables. Further, it double-counts repeated primes.
Goldoni [4] recently wrote a thesis on this general topic. His new results (Chapter 5) on the $h$ and $h^*$ invariants make it easier to use the results of Pleasants but do not extend them to cubic polynomials with fewer than 8 variables.
Of course I would be remiss in failing to mention the groundbreaking work of Heath-Brown [5], building on Friedlander & Iwaniec [6]. These results will no doubt clear the way for broader research, but so far have not been generalized.
So in short it appears that:
On the historical side, of course Fermat is responsible for the proof of the case $x^2+y^2$. I have references that say that Weber [7] and Schering [8] handled the case of (primitive) binary quadratic forms with nonsquare discriminants, but I haven't read the papers. Motohashi [9] proved that there are $\gg n/\log^2 n$ primes of the form $x^2+y^2+1$ up to $n$, apparently (?) the first such result with a constant term. He conjectured that the true number was
$$\frac{n}{(\log n)^{3/2}}\cdot\frac32\prod_{p\equiv3(4)}\left(1-\frac{1}{p^2}\right)^{-1/2}\left(1-\frac{1}{p(p-1)}\right)$$
but as far as I know the constant still has not been proved even for this special form.
Edit: Apparently Bredihin [10] proved the infinitude of primes of the form $x^2+y^2+1$ some years before Motohashi. He only gave a slight upper-bound on their density, though: $O(n/(\log n)^{1.042}).$ (Motohashi improved the exponent to 1.5 in a later paper.)
[1] Friedlander, J. and Iwaniec, H. (2010). Opera de Cribro. AMS.
[2] Pleasants, P. (1966). The representation of primes by cubic polynomials, Acta Arithmetica 12, pp. 23-44.
[3] Davenport, H. and Lewis, D. J. (1964). "Non-homogeneous cubic equations". Journal of the London Mathematical Society 39, pp. 657-671.
[4] Goldoni, L. (2010). Prime Numbers and Polynomials. Doctoral thesis, Università degli Studi di Trento.
[5] Heath-Brown, D. R. (2001). Primes represented by $x^3 + 2y^3$. Acta Mathematica 186, pp. 1-84; Wayback Machine.
[6] Friedlander, J. and Iwaniec, H. (1997). Using a parity-sensitive sieve to count prime values of a polynomial. Proceedings of the National Academy of Sciences 94, pp. 1054-1058.
[7] Weber, H. (1882). "Beweis des Satzes, dass, usw". Mathematische Annalen 20, pp. 301-329.
[8] Schering, E. (1909). "Beweis des Dirichletschen Satzes". Gesammelte mathematische Werke, Bd. 2, pp. 357-365.
[9] Motohashi, Y. (1969). On the distribution of prime numbers which are of the form $x^2+y^2+1$. Acta Arithmetica 16, pp. 351-364.
[10] Bredihin, B. M. (1963). Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood (Russian). Izv. Akad. Nauk. SSSR. Ser. Mat. 27, pp. 577-612.