Sums of five almost equal prime squares. II
[Part I, cf. Acta Arith. 77, 369-383 (1996; Zbl 0863.11067).] The main result obtained in this paper is that every integer \(N\equiv 5\pmod {24}\) can be represented as the sum of five squares of primes of almost equal magnitude. More precisely, the authors show that the five primes can all be chosen from the integral \([\sqrt{N/5}-U, \sqrt{N/5}+U]\) where \(U+ N^{12/25+ \varepsilon}\) for any \(\varepsilon> 0\). The key feature of this result lies in that \(U\) is smaller than \(\sqrt{N}\) by a power of \(N\). This is achieved by choosing the minor arcs considerably smaller so that for \(\alpha\) in the minor arcs, \[ \sum_{|n- \sqrt{N/5} |\leq U} \Lambda(n) e(n^2 \alpha)\ll U^2 N^{-\frac 12- \delta} \] for some \(\delta> 0\). Naturally, the widened major arcs are then more difficult to handle. For this, the authors follow the argument used in the paper [Monatsh. Math. 111, 147-169 (1991; Zbl 0719.11064)] of \textit{M. C. Liu} and the reviewer. Since the authors do not aim at a sharp estimate of the number of representations of \(N\), the influence of the possible Siegel's zero is avoided.
scientific article; zbMATH DE number 1207229