The exceptional set in the two prime squares and a prime problem (Q2505383)

The authors prove that almost all (with at most \(\ll N^ {5/12+\varepsilon}\) exceptions) of the numbers \(n \leq N\) with \(n \not\equiv -1,\bmod 3\) are expressible as a sum of a prime and two squares of primes. Their approach is via the Hardy-Littlewood method. In order to obtain their relatively sharp estimate of the size of the exceptional set the authors use two fairly recent additions to the available technology. On the minor arcs they use a method of \textit{T. D. Wooley} involving slim exceptional sets: see [Proc. Lond. Math. Soc. (3) Ser. 85, No. 1, 1--21 (2002; Zbl 1039.11066)]. On the major arcs, they invoke an approach of \textit{J. Y. Liu} and \textit{T. Zhan} [Sci. China, Ser. A 41, No. 7, 710--722 (1998; Zbl 0938.11048)] which avoids use of a possible Siegel zero or an appeal to the Deuring-Heilbronn phenomenon. The authors also use a lemma related to one appearing in a paper by \textit{M. C. Leung} and \textit{M. C. Liu} [Monatsh. Math. 115, No. 1-2, 133--169 (1993; Zbl 0779.11045)] on a different problem that also involves three prime variables.