Sums of nine almost equal prime cubes. (English) Zbl 1329.11110
L.-K. Hua [Q. J. Math., Oxf. Ser. 9, 68–80 (1938; Zbl 0018.29404; JFM 64.0131.02)] proved that all sufficiently large odd integers are sums of nine cubes of primes. X. M. Meng [J. Shandong Univ., Nat. Sci. Ed. 32, No. 4, 397–404 (1997; Zbl 0917.11038)] considered representations of the form
\[
2\nmid N= p^3_1+\cdots+ p^3_9;\;|p_i- (N/9)^{1/3}|\leq N^{(1/3)-\theta+\varepsilon};\;i= 1,\dots, 9;\;\varepsilon>0
\]
and showed that \(\theta= 1/198\) is available under the generalized Riemannian hypothesis. X. M. Meng [J. Shandong Univ., Nat. Sci. Ed. 37, No. 1, 31–38, 44 (2002; Zbl 1030.11054)] proved that \(\theta= 1/495\) is acceptable unconditionally. G. S. Lü [Acta Math. Sin., Chin. Ser. 49, No. 1, 195–204 (2006; Zbl 1215.11096)], G. S. Lü and Y. F. Xu [Acta Math. Hung. 116, No. 4, 309–326 (2007; Zbl 1164.11062)], T. Y. Li [Additive problems with prime numbers. Shandong: Shandong University (PhD Thesis) (2012)] improved the unconditional result to \(\theta= 2/555\), \(\theta= 1/198\), \(\theta= 1/90\), respectively.
In the paper under review, the author proves that \(\theta=1/51\) is acceptable unconditionally. The key point is the treatment of the integrals on the minor arcs, which combines an idea by L.-L. Zhao [Proc. Lond. Math. Soc. (3) 108, No. 6, 1593–1622 (2014; Zbl 1370.11116)] with new estimates for exponential sums over primes in short intervals obtained by A. V. Kumchev [in: Number theory. Arithmetic in Shangri-La. Proceedings of the 6th China-Japan seminar held at Shanghai Jiao Tong University, China, 2011. Hackensack, NJ: World Scientific. 116–131 (2013; Zbl 1368.11095)].
In the paper under review, the author proves that \(\theta=1/51\) is acceptable unconditionally. The key point is the treatment of the integrals on the minor arcs, which combines an idea by L.-L. Zhao [Proc. Lond. Math. Soc. (3) 108, No. 6, 1593–1622 (2014; Zbl 1370.11116)] with new estimates for exponential sums over primes in short intervals obtained by A. V. Kumchev [in: Number theory. Arithmetic in Shangri-La. Proceedings of the 6th China-Japan seminar held at Shanghai Jiao Tong University, China, 2011. Hackensack, NJ: World Scientific. 116–131 (2013; Zbl 1368.11095)].
Reviewer: Mihály Szalay (Budapest)
MSC:
| 11P55 | Applications of the Hardy-Littlewood method |
| 11P05 | Waring’s problem and variants |
| 11P32 | Goldbach-type theorems; other additive questions involving primes |
Citations:
Zbl 0018.29404; Zbl 0917.11038; Zbl 1030.11054; Zbl 1215.11096; Zbl 1164.11062; JFM 64.0131.02; Zbl 1370.11116; Zbl 1368.11095References:
| [1] | Hua L K. Some results in the additive prime number theory. Q J Math, 1938, 9(1): 68-80 · JFM 64.0131.02 · doi:10.1093/qmath/os-9.1.68 |
| [2] | Kumchev, A. V., On Weyl sums over primes in short intervals, No. 9, 116-131 (2012), Singapore · Zbl 1368.11095 |
| [3] | Li T Y. Additive Problems with Prime Numbers. Ph D Thesis. Shandong University, 2012 (in Chinese) |
| [4] | Liu J Y. An iterative method in the Waring-Goldbach problem. Chebyshevskii Sb, 2005, 5: 164-179 · Zbl 1145.11072 |
| [5] | Lü G S. Sums of nine almost equal prime cubes. Acta Math Sinica (Chin Ser), 2006, 49: 195-204 (in Chinese) · Zbl 1215.11096 |
| [6] | Lü G S, Xu Y F. Hua’s theorem with nine almost equal prime variables. Acta Math Hungar, 2007, 116(4): 309-326 · Zbl 1164.11062 · doi:10.1007/s10474-007-6041-6 |
| [7] | Meng X M. The Waring-Goldbach problems in short intervals. J Shandong Univ Nat Sci, 1997, 3: 164-225 |
| [8] | Meng X M. On sums of nine almost equal prime cubes. J Shandong Univ Nat Sci, 2002, 37: 31-37 · Zbl 1030.11054 |
| [9] | Vaughan R C. The Hardy-Littlewood Method. 2nd ed. Cambridge: Cambridge University Press, 1997 · Zbl 0868.11046 · doi:10.1017/CBO9780511470929 |
| [10] | Zhao L L. On the Waring-Goldbach problem for fourth and sixth powers. Proc Lond Math Soc, DOI: 10.1112/plms/pdt072 · Zbl 1370.11116 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
