Waring-Goldbach problem for unlike powers. (English) Zbl 1128.11043

Ren, Xiu Min; Tsang, Kai Man

Acta Math. Sin., Engl. Ser. 23, No. 2, 265-280 (2007).

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. 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 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, 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 J. Y. Liu and T. Zhan in [Sci. China, Ser. A 41, No. 7, 710–722 (1998; Zbl 0938.11048)], and with 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 K. Kawada and 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)].

Reviewer: George Greaves (Cardiff)

MSC:

11P32	Goldbach-type theorems; other additive questions involving primes
11P05	Waring’s problem and variants
11P55	Applications of the Hardy-Littlewood method

Keywords:

circle method; exceptional set; enlarged major arcs

Citations:

Zbl 0938.11048; Zbl 0991.11056; Zbl 0050.04003; Zbl 1035.11047; Zbl 0052.27901; Zbl 0940.11047; Zbl 1016.11046; Zbl 0044.03601

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