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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.

  1. 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.
  2. The paper allows only integers, not half-integers, as coefficients, and so does not give the number of primes, e.g., in A117112.
  3. 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.

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I'm finding a number of (slightly) older results by Peter Pleasants which may address the issue. I'll give more details as I find them. –  Charles Feb 14 '11 at 15:41
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Regarding 2., I am sure that retracing the arguments of [1] will allow half-integer constants with almost no trouble. Regarding 3., this is subtler; it's easy to write down what the asymptotic density ought to be, but the combinatorial sieve techniques employed by Iwaniec and other sieve theorists often (as in [1]) exploit positivity of various gnarly terms in order to discard them completely, which inevitably produces a lower bound rather than a precise asymptotic. For the most up-to-date material, you might try Friedlander and Iwaniec's book "Opera de Cribro". –  David Hansen Feb 14 '11 at 17:44
    
OK, my library has Opera de Cribro, I guess I'll check that out and see what it has. –  Charles Feb 14 '11 at 19:47
    
I'm having trouble with your "For example," partly because you define but never use $\Delta$ and $D$, partly because of confusion over $X$ and $x$, and mostly because $P(x,y)=x^2+y^2$ represents all primes congruent 1 mod 4, which should be asymptotic to $x/(2\log x)$. –  Gerry Myerson Feb 14 '11 at 22:40
    
I defined D and Delta so I could give the conditions under which the result holds. Unfortunately I neglected to give it! Let me edit. –  Charles Feb 15 '11 at 2:18

1 Answer 1

up vote 5 down vote accepted

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:

  • Nothing further is known about primes represented by quadratic polynomials.
  • Apart from $x^3+2y^3$, almost nothing is known about which primes are represented by cubic polynomials, though some results are known for how often such polynomials take on prime values provided $h^*$ and hence the number of variables is large enough.

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.

[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.

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Almost surely, Heath-Brown's result on $x^3+2y^3$ will generalize readily to $ax^3+by^3$. –  Greg Martin Sep 16 '11 at 21:38

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