On pairs of linear equations in three prime variables (Q1780295)

The authors study the solvability of the simultaneous Diophantine equations \[ a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1, \quad a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_3, \eqno{(*)} \] where \(a_{11}, \dots, a_{23}, b_1, b_2\) are integer parameters and the unknowns \(x_1, x_2, x_3\) are restricted to primes. Since the binary Goldbach problem is a special case of (\(*\)), one expects that it may be extremely difficult to solve the above problem for a particular choice of the coefficients. On the other hand, a result stating that (\(*\)) is solvable in the primes for almost all pairs \((b_1, b_2)\) should be within reach. Such a result was obtained by \textit{M. C. Liu} and \textit{K. M. Tsang} [J. Reine Angew. Math. 399, 109--136 (1989; Zbl 0667.10030)]. The main theorem of the paper is a quantitative form of the result of Liu and Tsang. Let \(W(X)\) be the set of pairs \((b_1, b_2)\) with \(1 \leq b_1, b_2 \leq X\) and such that the equations (\(*\)) have solutions over \(\mathbb R\) and over the finite field \(\mathbb F_p\) for every prime \(p\). Assume that the coefficients \(a_{11}, \dots, a_{23}\) are such that \[ \Delta_{12}\Delta_{13}\Delta_{23} \neq 0, \quad \gcd(\Delta_{12}, \Delta_{13}, \Delta_{23}) = 1, \qquad \Delta_{ij} = \det \begin{pmatrix} a_{1i} & a_{1j} \\ a_{2i} & a_{2j} \end{pmatrix}, \] and set \(B = \max\{ 3| a_{ij}| : i = 1, 2; \, j = 1, 2, 3 \}\). The authors prove that if \(X \gg B^{61890}\), there is an absolute constant \(\delta > 0\) such that the system (\(*\)) has solutions in prime \(x_1, x_2, x_3\) for all but \(O(X^2B^{-4 - \delta})\) pairs in \(W(X)\). The proof uses the circle method and explicit estimates for zero-free regions of Dirichlet \(L\)-functions.