A numerical bound for small prime solutions of some ternary linear equations. II (Q5947045)

Let \(a_1,a_2,a_3\) be nonzero coprime integers, and set \(A= \max|a_i|\). Let \(b\) be a positive integer and consider solutions of the equation NEWLINE\[NEWLINEa_1p_1+ a_2p_2+ a_3p_3= bNEWLINE\]NEWLINE in odd primes \(p_i\). Assuming that the natural local conditions are met it is shown that if the \(a_i\) are all positive then there are solutions as soon as \(b\gg A^{144}\). Moreover if the \(a_i\) are not all of the same sign there are always solutions with NEWLINE\[NEWLINE\max\{|a_i|p_i\}\ll b+A^{144}.NEWLINE\]NEWLINE In both cases the implied constant is effective. In Part I the first two authors had shown [Acta Arith. 84, 343--383 (1998; Zbl 0918.11053)] that these results hold with 144 replaced by the better exponent 45, but with an ineffective constant. NEWLINENEWLINENEWLINENaturally the proof hinges on a discussion of exceptional real zeros of \(L\)-functions (``Siegel zeros''). In addition there are a number of calculations involving zero density estimates.