Geodesic
On additive binary problems with semiprime numbers of a specific form
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part II, Tome 166 (2019), pp. 22-40.

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The paper is devoted to methods of solution of binary additive problems with semiprime numbers, which form sufficiently “rare” subsequences of the natural series. Additional conditions are imposed on these numbers; the main condition is belonging to so-called Vinogradov intervals. We solve two problems that are analogs to the Titchmarsh divisor problem; namely, based on the Vinogradov method of trigonometric sums, we obtain asymptotic formulas for the number of solutions to Diophantine equations with semiprome numbers of a specific form.
Keywords: binary additive problem, trigonometric sum, prime number, semiprime number, short interval. 
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N. A. Zinchenko. On additive binary problems with semiprime numbers of a specific form. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part II, Tome 166 (2019), pp. 22-40. http://geodesic.mathdoc.fr/item/INTO_2019_166_a2/

[1] Vinogradov A. I., “O plotnostnoi gipoteze dlya $L$-ryadov Dirikhle”, Izv. AN SSSR. Ser. mat., 29:4 (1965), 903–934 | Zbl

[2] Vinogradov I. M., “Nekotoroe obschee svoistvo raspredeleniya prostykh chisel”, Mat. sb., 7 (1940), 365–372

[3] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1971 | MR

[4] Vinogradov I. M., Osnovy teorii chisel, Lan, SPb.–M., 2004 | MR

[5] Gritsenko S. A., “Ob odnoi zadache I. M. Vinogradova”, Mat. zametki., 39:5 (1986), 625–640 | MR | Zbl

[6] Gritsenko S. A., “Ternarnaya problema Goldbakha i problema Goldbakha–Varinga s prostymi chislami, lezhaschimi v promezhutkakh spetsialnogo vida”, Usp. mat. nauk., 43:4 (262) (1988), 203–204 | MR

[7] Gritsenko S. A., “Tri additivnye zadachi”, Izv. RAN. Ser. mat., 56:6 (9262) (1992), 1198–1216 | Zbl

[8] Gritsenko S. A., Zinchenko N. A., “Ob otsenke odnoi trigonometricheskoi summy po prostym chislam”, Nauch. ved. BelGU. Ser. mat. fiz., 5 (148):30 (2013), 48–52

[9] Zinchenko N. A., “Binarnaya additivnaya zadacha s poluprostymi chislami spetsialnogo vida”, Chebyshev. sb., VI:2 (14) (2005), 145–162 | Zbl

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

[11] Linnik Yu. V., Dispersionnyi metod v binarnykh additivnykh zadachakh, Izd-vo LGU, L., 1961 | MR

[12] Khooli K., Primenenie metodov resheta v teorii chisel, Nauka, M., 1987

[13] Changa M. E., “Prostye chisla v spetsialnykh promezhutkakh i additivnye zadachi s takimi chislami”, Mat. zametki., 73:3 (2003), 423–-436 | DOI | MR | Zbl

[14] Bombieri E., “On the large sieve”, Mathematica., 12 (1965), 201–225 | MR | Zbl

[15] Balog A., Friedlander K. J., “A hybrid of theorems of Vinogradov and Piatetski-Shapiro”, Pac. J. Math., 156 (1992), 45–62 | DOI | MR | Zbl

[16] Titchmarsh E. C., “A divisor problem”, Rend. Circ. Mat. Palermo., 54 (1930), 414–429 | DOI | Zbl

[17] Tolev D. I., “On a theorem of Bombieri–Vinogradov type for prime numbers from a thin set”, Acta Arithm., 81:1 (1997), 57–68 | DOI | MR | Zbl