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.

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