Generalization of Goldbach’s ternary problem with almost equal terms
https://doi.org/10.22405/2226-8383-2023-24-4-264-298
Abstract
An asymptotic formula is obtained for the number of representations of a sufficiently large natural 𝑁 in the form 𝑏_1𝑝_1 + 𝑏_2𝑝_2 + 𝑏_3𝑝_3 = 𝑁 with the conditions
$$ |𝑏_𝑖 𝑝_𝑖 - N/3|⩽ 𝐻, 𝐻 ⩾ (𝑏_1*𝑏_2*𝑏_3)^(4/3)𝑁^2/3)(ln𝑁)&60, 𝑏_𝑖 ⩽ (ln𝑁)^(𝐵_𝑖), $$
where 𝑏_1, 𝑏_2, 𝑏_3, 𝑁 are pairwise coprime natural numbers, 𝐵_𝑖 — arbitrary fixed positive numbers
About the Authors
Zarullo Khusenovich Rakhmonov
National Academy of Sciences of Tajikistan, A. Dzhuraev Institute of Mathematics
Uzbekistan
Uzbekistan
doctor of physical and mathematical sciences, professor, academician of the National Academy of Sciences of Tajikistan, director of the A. Dzhuraev Institute of Mathematics
Ismail Allakov
Termez State University
Uzbekistan
Uzbekistan
doctor of physical and mathematical sciences, professor
Bahrom Kholtoraevich Abraev
Termez State University
Uzbekistan
Uzbekistan
References
1. Vinogradov, I. M., 1952, Izbrannye trudy. (Russian) [Selected works.], Izdat. Akad. Nauk SSSR, Moscow.
2. Vinogradov, I. M., & Karatsuba, A. A., 1986, “The method of trigonometric sums in number theory”, Proc. Steklov Inst. Math., vol. 168, pp. 3-30.
3. Haselgrove C. B., 1951, “Some theorems in the analitic theory of number”, J. London Math. Soc., vol. 26, pp. 273-277.
4. Pan Chengdong, Pan Chengbiao, 1990, “On estimations of trigonometric sums over primes in short intervals (III)”, Chinese Ann. of Math., vol. 2, pp. 138-147.
5. Zhan, T., 1991, “On the Representation of large odd integer as a sum three almost equal primes”, Acta Math Sinica. New ser., vol. 7, Is. 3, pp. 135 – 170.
6. Jutila, M., 1991, “Mean value etstimates for exponential sums with applications to 𝐿-functions”, Acta Arithmetica, vol. 57, Is. 2, pp. 93 – 114.
7. Jia Chao-hua, 1994, “Three primes theorem in a short interval (VII)//, Acta Mathematica Sinica. New ser., vol. 10, No 4, pp. 369 – 387.
8. Baker A., 1967, “On some diophantine inequalities involving primes”, J. Reine Angew. Math., vol. 228, pp. 166 – 181.
9. Allakov I., 2021, Otsenka trigonometricheskikh summ i ikh prilozheniya k resheniyu nekotorykh additivnykh zadach teorii chisel. (Russian) [Estimation of exponential sums and their applications to the solution of some additive problems in number theory], Termez. Ed. “Surkhan nashr”.
10. Rakhmonov Z. Kh., 2003, “Estermann’s ternary problem with almost equal summands”, Mathematical Notes, vol. 74, Is. 4, pp. 534-542.
11. Rakhmonov Z. Kh., & Mirzoabdugafurov K. I., 2008, “Waring’s problem for cubes with almost equal summands”, Doklady Akademii nauk Respubliki Tajikistan, vol. 51, no 2, pp. 83-86, (in Russian).
12. Rakhmonov Z. Kh., & Azamov A. Z., 2011, “An asymptotic formula in Waring’s problem for fourth powers with almost equal summands”, Doklady Akademii nauk Respubliki Tajikistan, vol. 54, no 3, pp. 34-42, (in Russian).
13. Rakhmonov, Z. Kh., 2014, “The Estermann cubic problem with almost equal summands“, Mathematical Notes, vol. 95, Is. 3-4, pp. 407–417. doi.org/10.1134/S0001434614030122.
14. Rakhmonov Z. Kh., & Nazrubloev N. N., Rakhimov A.O., 2015, “Short Weyl sums and their applications”, Chebyshevskii Sbornik, vol. 16, Is. 1, pp. 232–247, (in Russian).
15. Rakhmonov, Z. Kh., 2023, “Generalization of Waring’s problem for nine almost proportional cubes”, Chebyshevskii Sbornik, vol. 24, Is. ..., pp. ... – ..., (in Russian).
16. Rakhmonov, Z. Kh., 2021, “Estimates of short exponential sums with primes in major arcs”, Chebyshevskii Sbornik, vol. 22, Is. 4(81), pp. 199 – 223, (in Russian).
17. Rakhmonov, Z. Kh.,& Rakhmonov, F. Z., 2017, “Short Cubic Exponential Sums over Primes”, Proceedings of the Steklov Institute of Mathematics, vol. 296, pp. 211 – 233. doi.org/10.1134/S0081543817010175
18. Rakhmonov, Z. Kh., 1994, “Theorem on the mean value of 𝜓(𝑥, 𝜒) and its applications”, Russian Academy of Sciences. Izvestiya Mathematics, vol. 43, Is. 1, pp. 49 – 64. doi.org/10.1070/IM1994v043n01ABEH001558
19. Rahmonov, F. Z., 2011, “Estimate of quadratic trigonometric sums with prime numbers”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., no. 3, pp. 56 – 60.
20. Rakhmonov, Z. Kh.,& Rakhmonov, F. Z., 2014, “Sum of short exponential sums over prime numbers”, Doklady Mathematics, vol. 90, No 3, pp. 699–700. doi.org/10.1134/S1064562414070138.
21. Rakhmonov Z. Kh., Rahmonov F. Z., 2019, “Short cubic exponential sums with M¨obius function”, Chebyshevskii Sb., vol. 20. № 4(72). pp. 281 – 305, doi.org/10.22405/2226-8383-2018-20-4-281-305.
22. Karatsuba A. A., 1993, Basic analytic number theory, Springer-Verlag, Berlin, xiv+222 pp.
23. Davenport H., 1967, Multiplicative Number Theory, Markham Publishing Company, Chigago.
24. Arkhipov G. I. & Chubarikov V. N. & Karatsuba A. A. 2004. Trigonometric sums in number theory and analysis, Berlin–New-York: Walter de Gruyter, 554 p.
25. Prachar K., 1957, Primzahlverteilung, Springer-Verlag.
26. Rakhmonov Z. Kh., 1994, “Estimate of the density of the zeros of the Riemann zeta function”, Russian Math. Surveys, vol. 49, is. 2, pp. 168–169.
27. Vinogradov, I. M., 1976, Osobye varianty metoda trigonometricheskikh summ (Russian) [Special variants of the method of trigonometric sums], Izdat. “Nauka”, Moscow. 119 p.
28. Mardjhanashvili, K. K., 1939, “An estimate for an arithmetic sum”, Doklady Akad. Nauk SSSR, vol. 22, no 7, pp. 391–393.
29. Huxley М. N., 1972, “On the differences between consecutive primes”, Inventiones mathematicae, vol. 15, pp. 164 – 170.
30. Whittaker G. E.,& Watson T. N., 1915, A Course of Modern Analysis. Part 1. The processes of analysis. Part 2. The transcendental functions, Cambridge, Cambridge University Press, 620.
Review
For citations:
Rakhmonov Z.Kh., Allakov I., Abraev B.Kh. Generalization of Goldbach’s ternary problem with almost equal terms. Chebyshevskii Sbornik. 2023;24(4):264-298. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-264-298
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