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 |
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