Geodesic
New estimates for the exceptional set of the sum of two primes from an arithmetic progression
Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 21-41.

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

The paper studies the question of representing numbers as the sum of two primes from an arithmetic progression, that is, the binary Goldbach problem, when primes are taken from an arithmetic progression. New estimates are proved for the number of even natural numbers that are (possibly) not representable as a sum of two primes from an arithmetic progression and for a number representing a given natural number, as a sum of two primes from an arithmetic progression.
Keywords: The Dirichlet charakter, Dirichlet $L$-function, exceptional set, representation numbers,exceptional zero, exceptional nature, main member, remaining member. 
@article{CHEB_2022_23_2_a1,
     author = {I. Allakov and A. Sh. Safarov},
     title = {New estimates for the exceptional set of the sum of two primes from an arithmetic progression},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {21--41},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a1/}
}
TY  - JOUR
AU  - I. Allakov
AU  - A. Sh. Safarov
TI  - New estimates for the exceptional set of the sum of two primes from an arithmetic progression
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 21
EP  - 41
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a1/
LA  - ru
ID  - CHEB_2022_23_2_a1
ER  - 
%0 Journal Article
%A I. Allakov
%A A. Sh. Safarov
%T New estimates for the exceptional set of the sum of two primes from an arithmetic progression
%J Čebyševskij sbornik
%D 2022
%P 21-41
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a1/
%G ru
%F CHEB_2022_23_2_a1
I. Allakov; A. Sh. Safarov. New estimates for the exceptional set of the sum of two primes from an arithmetic progression. Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 21-41. http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a1/

[1] Allakov I., “O goldbakhovykh chislakh”, Chebyshevskii sbornik, 9:1 (2008), 13–17 | MR

[2] Vinogradov I.M., Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1980, 200 pp. | MR

[3] Vinogradov I.M., Osobye varianty metoda trigonometricheskikh summ, Nauka, M., 1976, 95 pp. | MR

[4] Khua-Lo-Gen, “Additivnaya teoriya prostykh chisel”, Tr. Matem. ins. im. V.A.Steklova, 1947, 3–179

[5] Khua-Lo-Gen, Metod trigonometricheskikh summ i ee prilozheniya v teoriya chisel, Mir, M., 1964, 18–19

[6] Sorput J.G.van.der, “Sur l'hypothese de Goldbach pour Presque tous les nombers pairs”, Acta arithm. Varshava, 2 (1937), 266–290

[7] Chudakov N.G., “O probleme Goldbakha”, Dokl. AN SSSR, 17 (1937), 331–334

[8] Estermann T., “On Goldbach's problem: Proof that almost all even positive integers are sums of two primes”, Proc.Lond. Math. Soc., 44:2 (1938), 307–314 | DOI | MR | Zbl

[9] Hardy G.H. and Littlewood J.E., “Some problems of partitio numerorum; III: On the expression of a number as sum of primes”, Acta Math., 1923, no. 44, 1–70 | DOI | MR

[10] Vaughan R.C., The Hardi-Littlewood method, v. 2, Second edition, Cambridge University Press, 1997, 176–256 | MR

[11] Lavrik A.F., “K binarnym problemam additivnoi teorii prostykh chisel v svyazi s metodom trigonometricheskikh summ I.M. Vinogradova”, Vestnik LGU, 1961, no. 13, 11–27 | Zbl

[12] Vaughan R.C., “On Goldbach' s problem”, Acta arithm., 22 (1972), 21–48 | DOI | MR | Zbl

[13] Montgomery H.L., Vaughan R.C., “The exceptional set in Goldbach's problem”, Acta arithm., 27 (1975), 353–370 | DOI | MR | Zbl

[14] Allakov I., Isklyuchitelnoe mnozhestvo summy dvukh prostykh, Dissertatsiya na soiskaniyu uchenoi stepeni kandidata fiz.-mat.nauk, LGU, L., 1983, 148 pp.

[15] Vinogradov A.I., “O binarnoi probleme Khardi-Littlvuda”, Acta arithm., 46 (1985), 33–56 | DOI | MR | Zbl

[16] Arkhipov G.I.,Chubarikov V.N., “Ob isklyuchitelnom mnozhestve v binarnoi probleme goldbakhova tipa”, Dokl. RAN, 387:3 (2002), 295–296 | MR | Zbl

[17] Plaksin V.A., “Ob odnom voprose Khua-Lo-Kena”, Mat. Zametki, 1990, no. 3(47), 78–90 | Zbl

[18] Allakov I., Reshenie nekotorykh additivnykh zadach teorii chisel analiticheskimi metodami, Ta'lim, Tashkent, 2012

[19] Allakov I., Safarov A.Sh., “Exceptional set of the sum of a prime number and a fixed degree of a prime number”, Russian Mathematics, 64 (2020), 8–21 | DOI | MR | Zbl

[20] Wu Fang, “On the solutions of the systems of linear equations with prime variables”, Acta Math. Sinica, 7 (1957), 102–121 | MR

[21] Liu M.C. , Tsang K.M., “Small prime solutions of linear equations”, Proc. Intern. Number. Th. Conf. (1987), Laval Uniyersity. Cand. Math. Soc., Berlin–New York, 1989, 595–624 | MR

[22] Chubarikov V.N., “Mnogomernye problemy teorii prostykh chisel”, Chebyshevskii sbornik, 4:12 (2011), 176–256 | MR

[23] Allakov I., “O predstavlenie chisel summoi dvukh prostykh chisel iz arifmeticheskoi progressii”, Izvestiya VUZov. “Matematika”, 2000, no. 8(459), 3–15 | Zbl

[24] Karatsuba A.A., Osnovy analiticheskoi teorii chisel, Nauka, M., 1983 | MR

[25] Davenport Harold, Multiplicative Number Theory, Second ed., Shringer-Verlag, New York–Heidelberg–Berlin, 1997 | MR

[26] Montgomery H.L. and Vaughan R.C., Multiplicative number theory, v. I, Classical theory, Cambridge University Press, New York, 2006 | MR

[27] Prakhar K., Raspredelenie prostykh chisel, Mir, M., 1967

[28] Gallagher P.X., “A large sieve density estimate near”, Inv. Math., 11 (1970), 329–339 | DOI | MR | Zbl

[29] Kolmogorov A.N., Fomin S.V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[30] Allakov I., Israilov M., “Otsenka trigonometricheskikh summ po prostym chislam v arifmeticheskoi progressii”, Doklady AN RUz., 1982, no. 4, 5–6 | MR | Zbl