Liu, Ming-Chit; Wang, Tianze A numerical bound for small prime solutions of some ternary linear equations. (English) Zbl 0918.11053 Acta Arith. 86, No. 4, 343-383 (1998). This paper is concerned with small solutions, in prime variables, of the equation \[ a_1p_1+ a_2p_2+ a_3p_3= b, \] subject to the natural necessary conditions. It was shown by M.-C. Liu and K.-M. Tsang [Small prime solutions of linear equations, in: Théorie des nombres, C. R. Conf. Int., Quebec/Can. 1987 (Walter de Gruyter, Berlin, 595-624 (1989; Zbl 0682.10043))], that there is an absolute constant \(B\) such that the smallest solution satisfies \[ \max_j| a_j| p_j\ll \max\{| b|,(| a_1|+| a_2|+ | a_3|)^B\}. \] In the present paper, it is shown that \(B= 45\) is admissible. Previously, K. Choi [Bull. Hong Kong Math. Soc. 1, 1-19 (1997)] had obtained \(B= 4190\).The proof uses the circle method, but requires detailed information about zeros of Dirichlet \(L\)-functions. This is obtained mainly from the reviewer’s work [D. R. Heath-Brown, Proc. Lond. Math. Soc. (3) 64, 265-338 (1992; Zbl 0739.11033)]. However, it is noteworthy that the authors improve the zero-free region to \[ \sigma\geq 1- 0.364/\log Q \] save for the usual exceptional zero. The reviewer’s result had a constant 0.348. Reviewer: D.R.Heath-Brown (Oxford) Cited in 3 ReviewsCited in 8 Documents MSC: 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method 11D04 Linear Diophantine equations 11N13 Primes in congruence classes 11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses Keywords:ternary linear equations; Goldbach problem; small solutions; circle method; zeros of Dirichlet \(L\)-functions; zero-free region Citations:Zbl 0682.10043; Zbl 0739.11033 PDFBibTeX XMLCite \textit{M.-C. Liu} and \textit{T. Wang}, Acta Arith. 86, No. 4, 343--383 (1998; Zbl 0918.11053) Full Text: DOI EuDML