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A numerical bound for small prime solutions of some ternary linear equations. (English) Zbl 0918.11053

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.

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
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