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The quest for repeated, infinitely frequent patterns in the primes, is certainly a very old one, and is often even a quest for the asymptotic frequency of these patterns, which is much harder. For instance proving that there are infinitely many primes is easy, finding a satisfactory answer for the question « How many (roughly) until N large ? » much less so. Inbetween lies the obtention of upper and lower bounds of the right shape, due by Chebyshev.

Here pattern stands for images of a system of polynomial forms. Given $\Psi=(\psi_1,\cdots,\psi_t)\in(\mathbb{Z}[X_1,\cdots,X_d])^t$ and a convex set $K=K_N\subset\mathbb{R}^d$ of volume increasing to infinity, we are thus interested in evaluating

$\displaystyle{\sum_{n\in K\cap\mathbb{Z}^d}\prod_{i=1}^t\Lambda(\psi_i(n))}$

for N large, where $\Lambda$ is the von Mangoldt function. The case where $d=1$ and $\psi_i(n)=n+b_i$ is the original question of Hardy and Littlewood, who proposed a tantalizing asymptotic behaviour but is still completely out of reach (even the question whether there are infinitely many $n$ such that $n,n+2$ are both primes is not settled). But the case where the system $\Psi$ is affine-linear (thus the polynomials are all of degree 1) and no two forms are affinely dependent was solved by Green and Tao in the celebrated article Linear equations in primes.

Similar results for more general polynomial forms are rare. We have to mention the famous work of Friedlander and Iwaniec yielding an asymptotic for the number of primes of the form $p=x^2+y^4$, where it appears that

$\displaystyle{\sum_{x\leq\sqrt{N},y\leq N^{1/4}}\Lambda(x^2+y^4)}\sim Cx^{3/4}$

for some constant $C>0$.

I have uploaded yesterday an article on the ArXiv which provides asymptotics of the same shape as the ones in the Hardy-Littlewood for a few exceptional polynomial patterns.  Thus for instance, I can tell how many arithmetic progressions of three primes smaller than N exist whose common difference is a sum of two squares – well not quite, because I have to weigh these arithmetic progressions by the number of representations of the common difference. Now this weight, giving a positive density to the set of sums of two squares, which is sparse, of density $N/\sqrt{\log N}$, just as the von Mangoldt function is a weight (mostly) on primes giving them a density, cannot be easily eliminated afterwards, in contrast to the von Mangoldt function (one can write for $n\leq N$ that $\Lambda(n)\sim 1_\mathbb{P}(n)\log n\sim 1_\mathbb{P}(n)\log N$).

More precisely, the result that naturally comes out concerning three term arithmetic progressions with common difference a sum of two squares is

$\displaystyle{\sum_{1\leq a\leq a+2d\leq N}\Lambda(a)\Lambda(a+d)\Lambda(a+2d)R(d)=\pi N^2/4\prod_p\beta_p+o(N^2)}$

where $R(n)=\mid\{(x,y)\in\mathbb{Z}^2\mid n=x^2+y^2\}\mid$ is the representation function and $\beta_p$ are some explicit constant which I don’t reproduce here. Moreover, we can generalise to other positive definite binary quadratic forms than this one, and there’s nothing special about length three: an asymptotic is available for any length. Here we notice that in some sense, the result is only seemingly polynomial, and truly rather linear: the polynomial nature of the pattern is enclosed in a linear input into the representation function of a quadratic form.

In fact, my article contains a more general result of which the one above is only a particular case. My work consisted in mingling the von Mangoldt function with the representation functions of quadratic forms, whose behaviour on linear systems have been already analysed respectively in by Green and Tao and Matthiesen. The idea is to consider sums of the form

$\displaystyle{\sum_{n\in\mathbb{Z}^d\cap K}\prod_{i=1}^tF_i(\psi_i(n))}$

where $F_i$ can be the von Mangoldt function or a representation function, and the $\psi_i$ are linear forms. The cohabitation of both types of functions went quite well. One delicate point was to eliminate biases modulo small primes of both types functions, an operation known as the $W$-trick. The difficulty is that while the value of the von Mangoldt function is more or less determined by the coprimality to small primes, it is not so for the representation function, which is also sensitive to the residue modulo large powers of small primes. Once this issue is adressed carefully, it is possible to majorize them by one and the same pseudorandom majorant, which opens the way to the application of the transference principle.

Similarly, the cohabitation between the von Mangoldt function and the divisor function is quite natural, yielding asymptotics for expressions such as $\sum\Lambda(n)\Lambda(n+ab)\Lambda(n+2ab)=\sum\Lambda(n)\Lambda(n+d)\Lambda(n+2d)\tau(d)$. This is reminiscent of the Titchmarsh divisor problem, the evaluation of $\sum_n\Lambda(n)\tau(n+a)$ or (almost equivalently) of $\sum_p\tau(p+a)$, but the latter expression involves a linear system of infinite complexity, and is thus altogether out of reach of my method, just as the twin primes or the basic Hardy-Littlewood conjecture.

One may hope to extend the generalised Hardy-Littlewood conjecture stated (and proven in the finite complexity case in the paper linked) by Green and Tao to polynomial systems. For example given a polynomial $\displaystyle{\phi \in \mathbb{Z}[x_1,\ldots,x_d]}$ and a bounded domain $K\subset \mathbb{R}^d$ (probably a convex body would be more reasonable), one may be interested in an asymptotic for

$\displaystyle{\lvert\{x=(x_1,\ldots,x_d)\in K\cap \mathbb{Z}^d\mid \phi(x)\text{ is prime}\}\rvert.}$

We will rather try to get an asymptotic of the form

$\displaystyle{\sum_{(x_1,\ldots,x_d)\in\mathbb{Z}^d\cap K}\Lambda(\phi(x_1,\ldots,x_d))=\beta_{\infty}\prod_p\beta_p+o(\beta_{\infty})}$

where $\beta_{\infty}$ is basically the number of points with positive integer coordinates in $K$, so hopefully $\beta_{\infty}=\text{Vol}(K\cap \phi^{-1}(\mathbb{R}_+))$, and the local factors take into account the obstructions or the absence of obstructions to primality modulo $p$. Recall that $\Lambda$ classically denotes the von Mangoldt function. There are only very few non-linear polynomials for which an asymptotic for the number of prime values is available. The easiest one is obviously $\phi(x,y)=x^2+y^2$. Indeed, the primes which are sum of two squares are the primes congruent to 1 modulo 4 (and also the prime 2, but it’s a single prime, so we don’t have to care), and they are represented 8 times each as a value of $\phi$. So

$\displaystyle{\sum_{x^2+y^2\leq N}\Lambda(x^2+y^2)\sim\sum_{n\leq N,n\equiv 1\text{ mod } 4}\Lambda(n)\sim 4N.}$

Now let’s check what the conjecture would say. Here $\beta_{\infty}=\text{Vol}(\{(x,y)\mid x^2+y^2\leq N\})\sim \pi N$. What about the $\beta_p$ ? They are supposed to be $\beta_p=\frac{p}{p-1}\mathbb{P}(\phi(x,y)\neq 0\text{ mod } p)$. Now it easy to check that $\mathbb{P}(\phi(x,y)\equiv 0\text{ mod } p)=\frac{2p-1}{p^2}$ if $p\equiv 1\text{ mod } 4$, and $\mathbb{P}(\phi(x,y)\equiv 0\text{ mod } p)=\frac{1}{p^2}$ otherwise. So $\beta_p=\frac{p-1}{p}$ in the former case, and $\beta_p=\frac{p+1}{p}$ in the latter case. Thus, $\prod_p\beta_p$ doesn’t converge absolutely, in contrast with the traditional Green-Tao situation, where $\beta_p=1+O(1/p^2)$… However, if we imagine that to each prime $p\equiv 1\text{ mod } 4$ corresponds a prime $p'\equiv 3 \text{ mod } 4$ with $p\approx p'$, we could compute the product of the $\beta_p$ by grouping the factors into pairs $\beta_p\times\beta_{p'}\approx 1-1/p^2$. A bit more precisely,

$\displaystyle{\prod_{p\equiv 1\text{ mod } 4,p\leq N}(1+1/p)\sim C\sqrt{N}}$

and

$\displaystyle{\prod_{p\equiv 1\text{ mod } 4,p\leq N}(1-1/p)\sim C'/\sqrt{N}}$

which implies that

$\displaystyle{\prod_{p\leq N}\beta_p}\sim CC'$

so this product is convergent, although not absolutely. I guess the constant $C'$ is $e^{-\gamma/2}$, see Mertens theorem, and that $C=e^{\gamma/2}\prod_{p\equiv 1\text{ mod } 4}(1-p^{-2})$. I don’t really know how to compute this convergent product. We quickly notice that $\beta_2=1$. If the product could be equal to $4/\pi$, which it can’t unfortunately, being smaller than 1, we would apparently get the correct result, but this remains a quite dodgy case, which should make one cautious about stating ambitious generalisations of the Hardy-Littlewood conjecture.