On Waring-Goldbach problem for fifth powers (Q2909032)

For \(s \geq 11\) and \(N \to \infty\), let \(E_s(N)\) denote the number of positive integers \(n \leq N\), with \(n \equiv s \pmod 2\), that cannot be expressed as sums of \(s\) fifth powers of primes. In 2005, the reviewer [Can. J. Math. 57, No. 2, 298-327 (2005; Zbl 1080.11070)] obtained the bounds NEWLINE\[NEWLINE E_s(N) \ll N^{1-\Delta_s} \qquad (11 \leq s \leq 20), NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Delta_s = \begin{cases} (2s - 21)/240 + \delta & \text{if } 11 \leq s \leq 18, \\ (2s - 21)/240 + 2/15 + \delta & \text{if } s = 19, 20, \end{cases} NEWLINE\]NEWLINE for some small \(\delta > 0\). The value of \(\delta\) in that result was left unspecified to allow the inclusion of the above bounds in a more general theorem. To determine its exact value, one would have to perform a careful analysis of the minor arc contribution in the underlying application of the Hardy--Littlewood circle method. In the paper under review, the author essentially undertakes that analysis and shows, much to this reviewer's surprise, that it leads to a small but not insignificant improvement on the bounds stated above. The author shows that the best possible choice of \(\delta\) is \(\approx 1/240 - 0.0036514\). This may seem like too small a number to bother about, but such a perception is deceiving. Not only is the author's theorem more transparent than the reviewer's old version, but also in the most interesting case -- namely, when \(s=11\), the old value \(\Delta_{11} = 1/240\) is significantly improved to \(\Delta_{11} = 1/214\).