Geodesic
Generalization of Goldbach's ternary problem with almost equal terms
Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 264-298.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $$ \left|b_ip_i-\frac{N}3\right|\le H, H\ge (b_1b_2b_3)^\frac43N^\frac23(\ln N)^{60}, b_i\le(\ln N)^{B_i}, $$ where $b_1$, $b_2$ $b_3$, $N$ are pairwise coprime natural numbers, $B_i$ — arbitrary fixed positive numbers.
Keywords: ternary Goldbach problem, almost equal terms, short exponential sum with primes, small neighborhood of centers of major arcs. 
@article{CHEB_2023_24_4_a15,
     author = {Z. Kh. Rakhmonov and I. Allakov and B. Kh. Abrayev},
     title = {Generalization of {Goldbach's} ternary problem with almost equal terms},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {264--298},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a15/}
}
TY  - JOUR
AU  - Z. Kh. Rakhmonov
AU  - I. Allakov
AU  - B. Kh. Abrayev
TI  - Generalization of Goldbach's ternary problem with almost equal terms
JO  - Čebyševskij sbornik
PY  - 2023
SP  - 264
EP  - 298
VL  - 24
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a15/
LA  - ru
ID  - CHEB_2023_24_4_a15
ER  - 
%0 Journal Article
%A Z. Kh. Rakhmonov
%A I. Allakov
%A B. Kh. Abrayev
%T Generalization of Goldbach's ternary problem with almost equal terms
%J Čebyševskij sbornik
%D 2023
%P 264-298
%V 24
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a15/
%G ru
%F CHEB_2023_24_4_a15
Z. Kh. Rakhmonov; I. Allakov; B. Kh. Abrayev. Generalization of Goldbach's ternary problem with almost equal terms. Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 264-298. http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a15/

[1] Vinogradov, I. M., Selected works, Izdat. Akad. Nauk SSSR, M., 1952 (Russian) | MR

[2] Vinogradov I. M., Karatsuba A. A., “The method of trigonometric sums in number theory”, Proc. Steklov Inst. Math., 168 (1986), 3–30 | MR | MR | Zbl

[3] Haselgrove C. B., “Some theorems in the analitic theory of number”, J. London Math. Soc., 26 (1951), 273–277 | DOI | MR | Zbl

[4] Pan Cheng-dong, Pan Cheng-biao, “On estimations of trigonometric sums over primes in short intervals (III)”, Chinese Ann. of Math., 2 (1990), 138–147 | MR | Zbl

[5] Zhan T., “On the Representation of large odd integer as a sum three almost equal primes”, Acta Math Sinica. New ser., 7:3 (1991), 135–170 | MR

[6] Jutila M., “Mean value etstimates for exponential sums with applications to $L$-functions”, Acta Arithmetica, 57:2 (1991), 93–114 | DOI | MR | Zbl

[7] Jia Chao-hua, “Three primes theorem in a short interval (VII)”, Acta Mathematica Sinica. New ser., 10:4 (1994), 369–387 | DOI | MR | Zbl

[8] Baker A., “On some diophantine inequalities involving primes”, J. Reine Angew. Math., 228 (1967), 166–181 | MR | Zbl

[9] Allakov I., Estimation of exponential sums and their applications to the solution of some additive problems in number theory, Ed. “Surkhan nashr”, Termez, 2021 (Russian)

[10] Rakhmonov Z. Kh., “Estermann's ternary problem with almost equal summands”, Mathematical Notes, 74:4 (2003), 534–542 | DOI | DOI | MR | Zbl

[11] Rakhmonov Z. Kh., Mirzoabdugafurov K. I., “Waring's problem for cubes with almost equal summands”, Doklady Akademii nauk Respubliki Tajikistan, 51:2 (2008), 83–86 (in Russian)

[12] Rakhmonov Z. Kh., Azamov A. Z., “An asymptotic formula in Waring's problem for fourth powers with almost equal summands”, Doklady Akademii nauk Respubliki Tajikistan, 54:3 (2011), 34–42 (in Russian) | MR

[13] Rakhmonov Z. Kh., “The Estermann cubic problem with almost equal summands”, Mathematical Notes, 95:3-4 (2014), 407–417 | DOI | DOI | MR | MR | Zbl

[14] Rakhmonov Z. Kh., Nazrubloev N. N., Rakhimov A.O., “Short Weyl sums and their applications”, Chebyshevskii Sbornik, 16:1 (2015), 232–247 (in Russian) | MR | Zbl

[15] Rakhmonov Z. Kh., “Generalization of Waring's problem for nine almost proportional cubes”, Chebyshevskii Sbornik, 24 (2023) (in Russian) | MR | Zbl

[16] Rakhmonov Z. Kh., “Estimates of short exponential sums with primes in major arcs”, Chebyshevskii Sbornik, 22:4(81) (2021), 199–223 (in Russian) | DOI

[17] Rakhmonov Z. Kh., Rakhmonov F. Z., “Short Cubic Exponential Sums over Primes”, Proceedings of the Steklov Institute of Mathematics, 296 (2017), 211–233 | DOI | DOI | MR | Zbl

[18] Rakhmonov Z. Kh., “Theorem on the mean value of $\psi(x,\chi)$ and its applications”, Russian Academy of Sciences. Izvestiya Mathematics, 43:1 (1994), 49–64 | DOI | MR | Zbl

[19] Rahmonov F. Z., “Estimate of quadratic trigonometric sums with prime numbers”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2011, no. 3, 56–60 | MR | Zbl

[20] Rakhmonov Z. Kh., Rakhmonov F. Z., “Sum of short exponential sums over prime numbers”, Doklady Mathematics, 90:3 (2014), 699–700 | DOI | DOI | MR | Zbl

[21] Rakhmonov Z. Kh., Rahmonov F. Z., “Short cubic exponential sums with Möbius function”, Chebyshevskii Sb., 20:4(72) (2019), 281–305 | DOI | MR | Zbl

[22] Karatsuba A. A., Basic analytic number theory, Springer-Verlag, Berlin, 1993, xiv+222 pp. | MR | MR | Zbl

[23] Davenport H., Multiplicative Number Theory, Markham Publishing Company, Chicago, 1967 | MR | MR | Zbl

[24] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric sums in number theory and analysis, Walter de Gruyter, Berlin-New-York, 2004, 554 pp. | MR | MR | Zbl

[25] Prachar K., Primzahlverteilung, Springer-Verlag, 1957 | MR | Zbl

[26] Rakhmonov Z. Kh., “Estimate of the density of the zeros of the Riemann zeta function”, Russian Math. Surveys, 49:2 (1994), 168–169 | DOI | MR | Zbl

[27] Mardjhanashvili K. K., “An estimate for an arithmetic sum”, Doklady Akad. Nauk SSSR, 22:7 (1939), 391–393

[28] Vinogradov I. M., Special variants of the method of trigonometric sums, Izdat. “Nauka”, M., 1976, 119 pp. (Russian)

[29] Huxley M. N., “On the differences between consecutive primes”, Inventiones mathematicae, 15 (1972), 164–170 | DOI | MR | Zbl

[30] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, Perev. s angl., v. 1, Osnovnye operatsii analiza, Izd. 2-e, Fizmatgiz, M., 1963, 342 pp.; Whittaker G. E., Watson T. N., A Course of Modern Analysis, v. 1, The processes of analysis, 1915 ; v. 2, The transcendental functions, Cambridge University Press, Cambridge, 620 pp. | MR