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

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)].

### 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
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### References:

 [1] Roth, K. F.: A problem in additive number theory. Proc. London Math. Soc., 53(2), 381–395 (1951) · Zbl 0044.03601 [2] Prachar, K.: Uber ein Problem vom Waring–Goldbach’schen. Typ. Monatsh. Math., 57, 66–74 (1953) · Zbl 0050.04003 [3] Prachar, K.: Uber ein Problem vom Waring–Goldbach’schen Typ II.. Monatsh. Math., 57, 113–116 (1953) · Zbl 0052.27901 [4] Bauer, C.: An improvement on a theorem of the Goldbach–Waring type. Rocky Mountain Journal of Mathematics, 31(4), 1–20 (2001) · Zbl 1035.11047 [5] Liu, J. Y., Zhan, T.: Sums of five almost equal prime squares (II). Sci. in China., 41, 710–722 (1998) · Zbl 0938.11048 [6] Liu, J. Y., Liu, M. C., Zhan, T.: Squares of primes and powers of 2. Monatsh. Math., 128, 283–313 (1999) · Zbl 0940.11047 [7] Ren, X.: The exceptional set in Roth’s theorem concerning a cube and three cubes of primes. Quart. J. Math., 52, 107–126 (2001) · Zbl 0991.11056 [8] Kawada, K., Wooley, T. D.: On the Waring–Goldbach Problem for Fourth and Fifth Powers. Proc. London Math. Soc., 83(3), 1–50 (2001) · Zbl 1016.11046 [9] Vinogradov, I. M.: Elements of Number Theory, Dover Publications, 1954 · Zbl 0057.28201 [10] Davenport, H.: Multiplicative Number Theory, 2nd ed., Springer, Berlin, 1980 · Zbl 0453.10002 [11] Huxley, M. N.: Large values of Dirichlet polynomials (III). Acta Arith., 26, 435–444 (1975) · Zbl 0268.10026 [12] Jutila, M.: On Linnik’s constant. Math. Scand., 41, 45–62 (1977) · Zbl 0363.10026 [13] Heath–Brown, D. R.: The density of zeros of Dirichlet’s L–functions. Can. J. Math., 31(2), 231–240 (1979) · Zbl 0396.10029 [14] Titchmarsh, E. C.: The theory of the Riemann zeta–function, 2nd ed., University Press, Oxford, 1986 · Zbl 0601.10026 [15] Prachar, K.: Primzahlverteilung, Springer, Berlin 1957 [16] Hua, L. K.: Additive theory of prime numbers, Science Press, Beijing 1957; English Version, Amer. Math. Soc., Rhode Island, 1965
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