Small prime solutions of linear ternary equations (Q2717632)

Let \(a_1\), \(a_2\), \(a_3\), \(b\) be integers satisfying certain natural conditions, including the requirement that \(a_1\), \(a_2\), \(a_3\) are not all of the same sign. It is proved that the equation NEWLINE\[NEWLINEa_1 p_1+ a_2 p_2+ a_3 p_3= bNEWLINE\]NEWLINE has a solution in primes \(p_i\) satisfying NEWLINE\[NEWLINE\max|a_i|p_i\ll \max \{|b|,\;(\max\{|a_1|, |a_2|, |a_3|\}^{38})\}.NEWLINE\]NEWLINE Previously, \textit{M.-C. Liu} and \textit{T. Wang} [Acta Arith. 86, 343-383 (1998; Zbl 0918.11053)] had established a similar result with the exponent 38 replaced by 45. NEWLINENEWLINENEWLINEThe proof uses explicit zero-free regions and explicit zero-density estimates for Dirichlet \(L\)-functions. A fair amount of computation is therefore involved in the argument.