Distribution of integers that are sums of three squares of primes (Q2717627)

It is conjectured that every sufficiently large integer \(n\equiv 3\pmod {24}\) for which \(5\nmid n\) should be the sum of the squares of 3 primes. Let \(E(N)\) be the number of such integers \(n\leq N\) which cannot be so represented. It was shown by \textit{L.-K. Hua} [Q. J. Math., Oxf. 9, 68-80 (1938; Zbl 0018.29404)] that \(E(N)= o(N)\). This was improved by other authors, most recently by \textit{C. Bauer, M.-C. Liu} and \textit{T. Zhan} [J. Number Theory 85, 336-359 (2000; Zbl 0961.11034)], who showed that \(E(N)\ll N^\theta\) for any \(\theta> 77/80= 0.9625\). The present paper improves this further to allow any \(\theta> 47/50= 0.94\). The circle method is used, as in Bauer et al., but wider major arcs are employed. The principal difficulty in the proof is consequently in handling these wider major arcs.