Generalization of Waring’s problem for nine almost proportional cubes
https://doi.org/10.22405/2226-8383-2023-24-3-71-94
Abstract
An asymptotic formula is obtained for the number of representations of a sufficiently large natural 𝑁 as a sum of nine cubes of natural numbers 𝑥𝑖, 𝑖 = 1, 9, satisfying the conditions
$$|(𝑥_𝑖)^3− 𝜇𝑖𝑁| ⩽ 𝐻, 𝜇1 + . . . + 𝜇9 = 1 𝐻 ⩾ 𝑁^)1−(1/30)+𝜀), $$
where 𝜇1, . . . , 𝜇9 — positive fixed numbers. This result is a strengthening of E.M.Wright’s theorem.
About the Author
Zarullo Khusenovich Rakhmonov
National Academy of Sciences of Tajikistan, A. Dzhuraev
Institute of Mathematics
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, professor, Academician of the National Academy of Sciences of Tajikistan, director of the A. Dzhuraev Institute of Mathematics
References
1. Waring E. M., 1770, Meditationes algebraicae , Cambridge.
2. Hilbert D., 1909, “Beweis f¨ur die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl
3. n-ter Potenzen (Waringsches Problem)”, Mathematische Annalen, vol. 67, pp. 281—300.
4. Hardy G. H., & Littlwood J.E., 1920, “Some problems of ”Partitio Numerorum”. I: A
5. new solution of Waring’s problem”, Nachrichten von der Gesellschaft der Wissenschaften zu
6. G¨ottingen. Mathematisch-Physikalische Klasse, pp. 33-54.
7. Vinogradov I. M., 1952, Izbrannye trudy. (Russian) [Selected works.], Izdat. Akad. Nauk SSSR, Moscow, 436 pp., (in Russian).
8. Vinogradov I.M., 1934, “Novoe reshenie problemy Varinga”, Doklady Akademii nauk, vol. 2,
9. pp. 337–341, (in Russian).
10. Vinogradov I. M., 1959, “On an upper bound for 𝐺(𝑛)”, Izv. Akad. Nauk SSSR Ser. Mat.,
11. vol. 23, no 5, pp. 637–642, (in Russian).
12. Karatsuba A. A., 1986, “On the function 𝐺(𝑛) in Waring’s problem”, Math. USSR-Izv., vol. 27,
13. no 2, pp. 239–249.
14. Wooley T. D., 1992, “Large improvements in Waring’s problem”, Ann of Math., vol. (2)135,
15. no 1. pp. 131-164.
16. Davenport H., 1939, “On Waring’s Problem for Fourth Powers”, Annals of Mathematics Second Series, vol. 40, no. 4, pp. 731-747.
17. Linnik Yu. V.,1942, “On the representation of large numbers as sums of seven cubes”,
18. C. R. (Dokl.) Acad. Sci. URSS, n. Ser., vol. 35, pp. 162.
19. Vaughan, R. C., 1986, “On Waring’s problem for cubes”, J. Reine Angew. Math., vol. 365,
20. pp. 122-170.
21. Wright E. M., 1934, “Proportionality conditions in Waring’s problem”, Mathematische
22. Zeitschrift, vol. 38, pp. 730-746.
23. Wright E. M., 1933, “An extension of Waring’s problem”, Philosophical Transactions of the
24. Royal Society of London, Series A, vol. 232. pp. 1-26.
25. Rakhmonov, Z. Kh., 2014, “The Estermann cubic problem with almost equal summands“,
26. Mathematical Notes, vol. 95, Is. 3-4, pp. 407–417. doi.org/10.1134/S0001434614030122.
27. Rakhmonov Z. Kh., & Nazrubloev N. N., Rakhimov A.O., 2015, “Short Weyl sums and their
28. applications”, Chebyshevskii Sbornik, vol. 16, Is. 1, pp. 232–247, (in Russian).
29. Rakhmonov Z. Kh., 2003, “Estermann’s ternary problem with almost equal summands”,
30. Mathematical Notes, vol. 74, Is. 4, pp. 534-542. doi.org/10.1023/A:1026199928464.
31. Rakhmonov Z. Kh., 2013, “Short Weyl sums”, Uchenyye zapiski Orlovskogo universiteta. Seriya yestestvennyye, tekhnicheskiye i meditsinskiye nauki, no. 6, part 2, pp. 194–203, (in Russian).
32. Rakhmonov Z. Kh., & Azamov A.Z., Nazrubloev N. N., 2018, “Of short Weyl’s exponential sum in minor arcs”, Doklady Akademii nauk Respubliki Tajikistan, vol. 61, no 7-8, pp. 609-614, (in Russian).
33. Rakhmonov Z. Kh., & Mirzoabdugafurov K. I., 2008, “Waring’s problem for cubes with almost
34. equal summands”, Doklady Akademii nauk Respubliki Tajikistan, vol. 51, no 2, pp. 83-86, (in
35. Russian).
36. Rakhmonov Z. Kh., & Azamov A.Z., 2011, “An asymptotic formula in Waring’s problem for
37. fourth powers with almost equal summands”, Doklady Akademii nauk Respubliki Tajikistan,
38. vol. 54, no 3, pp. 34-42, (in Russian).
39. Rakhmonov Z. Kh., & Nazrubloev N. N., 2014, “Waring’s problem for fifth powers with almost
40. equal summands”, Doklady Akademii nauk Respubliki Tajikistan, vol. 57, no 11-12, pp. 823-830,
41. (in Russian).
42. Rakhmonov F. Z., & Rakhimov A. O., 2015, “On an additive problem with almost equal
43. summands”, Issledovaniya po algebre, teorii chisel, funktsional’nomu analizu i smezhnym
44. voprosam. Izdatel’stvo: Saratovskiy natsional’nyy issledovatel’skiy gosudarstvennyy universitet
45. im. N.G. Chernyshevskogo, ISSN: 1810-4134, no 8, pp. 87–89, (in Russian).
46. Arkhipov G. I. & Chubarikov V. N. & Karatsuba A. A. 2004. Trigonometric sums in number
47. theory and analysis, Berlin–New-York: Walter de Gruyter, 554 p.
48. Whittaker G. E., Watson T. N., 1915, A Course of Modern Analysis. Part 1. The processes of
49. analysis. Part 2. The transcendental functions, Cambridge, Cambridge University Press, 620.
50. Vaughan R. C., 1981. The Hardy-Littlewood method, Cambridge Tracts in Mathematics, vol.
51. , Cambridge University Press, Cambridge, 172 p.
Review
For citations:
Rakhmonov Z.Kh. Generalization of Waring’s problem for nine almost proportional cubes. Chebyshevskii Sbornik. 2023;24(3):71-94. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-3-71-94
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