Conditional bounds for small prime solutions of linear equations
Let \(a_ 1,a_ 2,a_ 3,b\) be integers satisfying \(b\equiv a_ 1+a_ 2+a_ 3\pmod 2\) and \((a_ 1,a_ 2,a_ 3)=(b,a_ i,a_ j)=1\) for \(1\leq i<j\leq 3\). In earlier papers [J. Reine Angew. Math. 399, 109-136 (1989; Zbl 0667.10030) and Monatsh. Math. 111, 147-169 (1991; Zbl 0719.11064)] the second and third author considered solutions to the equation \[ a_ 1p_ 1+a_ 2p_ 2+a_ 3p_ 3=b \tag{*} \] in primes \(p_ 1,p_ 2,p_ 3\). They proved that there are constants \(A_ 1,A_ 2\) such that (i) if \(a_ 1,a_ 2,a_ 3>0\) and \(b\geq (3\max\{a_ 1,a_ 2,a_ 3\})^{A_ 1}\), then \((*)\) is soluble; (ii) if \(a_ 1,a_ 2,a_ 3\) are not of the same sign, then \((*)\) has solutions satisfying \[ \max\{p_ 1,p_ 2,p_ 3\}\leq 3| b|+(3\max\{| a_ 1|,| a_ 2|,| a_ 3|\})^{A_ 2}. \] Observe that (i) includes the classical three primes theorem of Vinogradov, while (ii) can be applied to show that if \(q,\ell\) are coprime, then the least prime \(p\equiv\ell\pmod q\) satisfies \(p\ll q^{A_ 2}\). An estimate on \(A_ 2\) will therefore give an estimate for Linnik's constant, and the first author has shown that \(A_ 1,A_ 2<4191\). In the present paper the authors replace (i) and (ii) by very much more precise estimates established under the assumption of the Generalised Riemann Hypothesis. In particular, the conditional results give \(A_ 1<5+\varepsilon\), \(A_ 2<4+\varepsilon\) for any \(\varepsilon>0\), and also, for the least prime in arithmetic progression problem, the bound \(p\ll q^ 2\log^{10}q\), which is only slightly weaker than the well known bound of \(q^ 2\log^ 2q\) obtainable directly under GRH. Appropriate generating functions have to be introduced so that the problem is brought under the circle method and the relevance of GRH. The new result is based heavily on the ingenious evaluation of a certain generalised Gauss sum.
small prime solutions of linear equations
scientific article; zbMATH DE number 60075
scientific article