Waring-Goldbach problem for unlike powers
The problem considered here is that of representing a positive integer~\(n\) as a sum \(p_2^ 2+p_3^ 3+p_4^ 4+p_5^ 5\) of prime powers. Let \(E(N)\) denote the number of ``exceptional'' even numbers \(n \leq N\) that are not expressible in this way. \textit{K.~Prachar} [Monatsh. Math. 57, 66--74 (1953; Zbl 0050.04003)] proved that \(E(N) \ll N/\log^ cN\) whenever \(c < {30 \over 47}\), and more recently \textit{C.~Bauer} [Rocky Mt. J. Math. 1, No. 4, 1151--1170 (2001; Zbl 1035.11047)] showed that \(E(N) \ll N^ {1-\delta}\) for some certain \(\delta\) slightly better than \(1\over2742\). The authors now establish the result with \(\delta={1\over66}\). In the problem in which the first power~\(p_1\) of a prime~\(p_1\) is added to the summand, \textit{K. Prachar} [Monatsh.\ Math.~57, 113--116 (1953; Zbl 0052.27901)] proved that all sufficiently large odd integers are representable, a fact which the authors recover as a corollary of their main result. The proof uses the circle method, using a approach used by \textit{J. Y. Liu} and \textit{T. Zhan} in [Sci. China, Ser. A 41, No. 7, 710--722 (1998; Zbl 0938.11048)], and with \textit{M. C. Liu} in [Monatsh. Math. 128, No. 4, 283--313 (1999; Zbl 0940.11047)], which allows enlarged major arcs in a way that does not invoke the Deuring--Heilbronn phenomenon or an appeal to the properties of a possible Siegel zero. This approach appears in a number of other papers in the recent literature, one of them [Q. J. Math. 52, No. 1, 107--126 (2001; Zbl 0991.11056)] by the first-named author of the paper currently under review. The authors are now able to make use of classical zero-density estimates for \(L\)-functions, in a way that they make more efficient by a certain iterative treatment. On the minor arcs, that authors use an estimate of \textit{K. Kawada} and \textit{T. D. Wooley} from a paper [Proc. Lond. Math. Soc. (3) 83, No. 1, 1--50 (2001; Zbl 1016.11046)] on the Waring-Goldbach problem for fourth and fifth powers. They also use a lemma of K. F. Roth from his paper on sums of mixed powers of natural numbers [Proc. Lond. Math. Soc. (2) 53, 381--395 (1951; Zbl 0044.03601)].
scientific article; zbMATH DE number 5162303
scientific article