Geodesic
A~sharp estimate for the number of solutions of a~system of Diophantine equations
Izvestiya. Mathematics , Tome 13 (1979) no. 3, pp. 461-497.

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A sharp estimate is obtained for the average of $|\Gamma|^{2k}$, where $\Gamma$ is an $r$-fold Weyl sum, valid for suitably large $k$. Bibliography: 9 titles. 
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G. I. Arkhipov; A. A. Karatsuba; V. N. Chubarikov. A~sharp estimate for the number of solutions of a~system of Diophantine equations. Izvestiya. Mathematics , Tome 13 (1979) no. 3, pp. 461-497. http://geodesic.mathdoc.fr/item/IM2_1979_13_3_a0/

[1] Vinogradov I. M., Izbrannye trudy, M., 1952 | MR

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

[3] Arkhipov G. I., Chubarikov V. N., “Kratnye trigonometricheskie summy”, Izv. AN SSSR. Ser. matem., 40 (1976), 209–220 | Zbl

[4] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., “Verkhnyaya granitsa modulya kratnoi trigonometricheskoi summy”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 143, 1977, 3–31 | Zbl

[5] Karatsuba A. A., “Srednee znachenie modulya trigonometricheskoi summy”, Izv. AN SSSR. Ser. matem., 37 (1973), 1203–1227 | Zbl

[6] Arkhipov G. I., Karatsuba A. A., “Novaya otsenka integrala I. M. Vinogradova”, Izv. AN SSSR. Ser. matem., 42 (1978), 751–762 | MR | Zbl

[7] Arkhipov G. I., “Teorema o srednem znachenii modulya kratnoi trigonometricheskoi summy”, Matem. zametki, 17:1 (1975), 143–153 | MR | Zbl

[8] Karatsuba A. A., “Teoremy o srednem i polnye trigonometricheskie summy”, Izv. AN SSSR. Ser. matem., 30 (1966), 183–206 | Zbl

[9] Arkhipov G. I., “Otsenki dvoinykh trigonometricheskikh summ G. Veilya”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 142, 1976, 46–66 | MR | Zbl