The quadratic form in nine prime variables (Q2828019)

The goal of this paper is to establish a local-global principle for the problem of representing an integer by a quadratic form in at least 9 prime variables. Let \(f(x_1,\ldots, x_n)\) be a regular indefinite integral quadratic form with \(n\geq 9\). Assume that the equation \(f(x_1,\ldots, x_n)=t\) has a solution in \(\mathbb{R}_{>0}\) and in the \(p\)-adic units for all primes \(p\). Then the equation \(f(x_1,\ldots, x_n)=t\) has a solution with all of the \(x_1,\ldots, x_n\) prime as soon as \(n\geq 9\). NEWLINENEWLINEPreviously \textit{J. Liu} [Monatsh. Math. 164, No. 4, 439--465 (2011; Zbl 1292.11073)] proved that under the same local assumptions the set of integer solutions with all \(x_1,\ldots, x_n\) prime is Zariski-dense in the affine quadric defined by \(f(x_1,\ldots, x_n)=t\) for certain classes of quadratic forms as soon as \(n\geq 10\). Work of \textit{B. Cook} and \textit{Á. Magyar} [Invent. Math. 198, No. 3, 701--737 (2014; Zbl 1360.11063)] on rather general systems of polynomial equations leads to similar results for all regular quadratic forms in at least \(21\) variables. This could be improved by work of \textit{E. Keil} [``Translation invariant quadratic forms in dense sets'', Preprint, \url{arXiv:1308.6680}] on solutions to \(f(x_1,\ldots, x_n)=t\) with variables lying in a set of positive upper density, to at least \(17\) variables. Note that Keil's work is also in parts inspired by the work of Liu. For a diagonal quadric \textit{L.-K. Hua}'s work [Q. J. Math., Oxf. Ser. 9, 68--80 (1938; Zbl 0018.29404; JFM 64.0131.02)] establishes Zariski-density of solutions with prime coordinates as soon as there are no bad primes and \(n\geq 5\). NEWLINENEWLINEThe main result in the paper under review, as well as in previous work, is established in studying the counting function NEWLINE\[NEWLINE N_{f,t}(X)= \sum_{{1\leq x_1,\ldots, x_n\leq X}\atop{f(x_1,\ldots, x_n)=t}}\prod_{j=1}^n \Lambda (x_j),NEWLINE\]NEWLINE where \(\Lambda\) is the von Mangoldt function. The authors make use of the Hardy-Littlewood circle method and establish an asymptotic formula of the shape NEWLINE\[NEWLINEN_{f,t}(X)=\mathfrak{S}(f,t)\mathcal{J}_{f,t}(X)+O_K(X^{n-2}\log^{-K}X),NEWLINE\]NEWLINE for \(K\) arbitrarily large. The major arc asymptotic is already deduced under the milder assumption that the rank of the quadratic matrix \(A\) associated to \(f\) is at least \(5\). As in previous work in the subject, the proof for the minor arc bound proceeds via a case distinction which depends on the so called off-diagonal rank of \(A\).