On multiple rational trigonometric sums over a field of algebraic numbers

V. N. Chubarikov

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 306-323.

Résumé

The paper describes the basic properties of polynomial comparisons modulo an ideal in the ring of integers of an algebraic number field, estimates of total rational trigonometric sums from a polynomial over an algebraic field are found, estimates of sums of Dirichlet characters modulo the degree of a prime ideal in an algebraic field are obtained, estimates of multiples of total rational trigonometric sums from polynomials over an algebraic field are given.

Détail

Keywords: trigonometric sums, I. M. Vinogradov method, Hua Lo-ken method, ring of integers in an algebraic number field, complete rational trigonometric sums over an algebraic number field, Dirichlet characters in algebraic number fields, A. G. Postnikov formula for Dirichlet characters in an algebraic field.

Bibliographie

[1] Vinogradov I. M., “Novyi sposob dlya polucheniya asimptoticheskikh vyrazhenii arifmeticheskikh funktsii”, Izv. RAN, 41:16 (1917), 1347–1378

[2] Vinogradov I. M., “O srednem znachenii chisla klassov chisto korennykh form otritsatelnogo opredelitelya”, Soobsch. Khark. mat. o-va, 1:1–2 (1918), 10–38

[3] Vinogradov I. M., “Sur la distribution des résidues et des non-résidues des puissances”, Zhurn. Fiz.-mat. o-va pri Perm. un-te, 1918, no. 1, 94–98

[4] Vinogradov I. M., “Elementarnoe dokazatelstvo odnoi obschei teoremy analiticheskoi teorii chisel”, Izv. RAN, 19:16–17 (1925), 785–796 (Rez. na frants. yaz.)

[5] Vinogradov I. M., “O raspredelenii indeksov”, Dokl. AN SSSR-A, 1926, no. 4, 73–76

[6] Vinogradov I. M., “On the bound of the least non-residues of $n$-th powers”, Trans. Am. math. Soc., 29:1 (1927), 218–226 | MR

[7] Winogradow J. M., “Sur un théorème général de Waring”, Matem. sbornik, 31 (1924), 490–507

[8] Vinogradov I. M., “K voprosu o raspredelenii drobnykh dolei znachenii funktsii odnogo peremennogo”, Zhurn. Leningr. fiz.-mat. o-va, 1:1 (1926), 56–65 (Rez. na frants. yaz.)

[9] Vinogradov I. M., “O raspredelenii drobnykh dolei znachenii funktsii dvukh peremennykh”, Izv. Leningr. politekhn. in-ta, 30 (1927), 31–52 (Rez. na frants. yaz.)

[10] Vinogradov I. M., “O teoreme Varinga”, Izv. AN SSSR. OFMN, 1928, no. 4, 393–400

[11] Vinogradov I. M., “Novoe reshenie problemy Varinga”, Dokl. AN SSSR, 2:6 (1934), 337–341 | Zbl

[12] Vinogradov I. M., “O nekotorykh novykh problemakh teorii chisel”, Dokl. AN SSSR, 3:1 (1934), 1–6 | MR | Zbl

[13] Vinogradov I. M., “Novaya otsenka $G(n)$ v probleme Varinga”, Dokl. AN SSSR, 4:5-6 (1934), 249–253 | Zbl

[14] Vinogradov I. M., “O verkhnei granitse $G(n)$ v probleme Varinga”, Izv. AN SSSR. OMEN, 1934, no. 10, 1455–1469

[15] Vinogradov I. M., “Novye otsenki summ Veilya”, Dokl. AN SSSR, 3:5 (1935), 195–198 | Zbl

[16] Vinogradov I. M., “Predstavlenie nechetnogo chisla summoi trekh prostykh chisel”, Dokl. AN SSSR, 15:6-7 (1937), 291–294

[17] Vinogradov I. M., “Otsenki nekotorykh prosteishikh trigonometricheskikh summ s prostymi chislami”, Izv. AN SSSR. Ser. mat., 1939, no. 4, 371–398 | Zbl

[18] Vinogradov I. M., “Nekotorye problemy analiticheskoi teorii chisel”, Trudy Tretego Vsesoyuznogo matematicheskogo s'ezda, Obzornye doklady (Moskva, iyun–iyul 1956 g.), v. 3, Izd-vo AN SSSR, M., 1958, 3–13

[19] Vinogradov I. M., “K voprosu o raspredelenii drobnykh chastei znachenii mnogochlena”, Izv. AN SSSR. Ser. mat., 25:6 (1961), 749–754 | Zbl

[20] Vinogradov I. M., Karatsuba A. A., “Metod trigonometricheskikh summ v teorii chisel”, Trudy matem. in-ta AN SSSR, 168, 1984, 4–30 | MR | Zbl

[21] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1980, 144 pp. | MR

[22] Vinogradov I. M., Osobye varianty metoda trigonometricheskikh summ, Nauka, M., 1976, 120 pp. | MR

[23] Vinogradov I. M., Osnovy teorii chisel, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.-Izhevsk, 2005, 176 pp.

[24] Vinogradov I. M., Selected Works, Springer-Verlag, N.Y.– Heidelberg–Berlin, 1985, 401 pp. | MR | Zbl

[25] Gauss K. F., Disquisitiones arithmeticae, Fleischer, Leipzig, 1801

[26] Hardy G. H., Littlwood J. E., “Some problems of “Partitio Numerorum”:III. On the expression of a number as a sum of primes”, Acta Math., 44 (1923), 1–70 | DOI | MR

[27] Hardy G. H., Littlwood J. E., “Some problems of “Partitio Numerorum”: VI. Further researches in Waring's problem”, Math. Z., 23 (1925), 1–37 | DOI | MR

[28] Landau E., Vorlesungen über Zahlentheorie, v. I, S. Hirzel, Leipzig, 1927

[29] Hua Loo-Keng, Selected Papers, N.Y.–Heidelberg–Berlin, 1983, 888 pp. | MR | Zbl

[30] Hua Loo-Keng, “Some results in the additive prime number theory”, Quart. J. Math. Oxford, 9 (1938), 68–80 | DOI | MR | Zbl

[31] Postnikov A. G., “O summe kharakterov po modulyu, ravnomu stepeni prostogo chisla”, Izv. AN SSSR, ser. matem., 19:1 (1955), 11–16 | Zbl

[32] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., Teoriya kratnykh trigonometricheskikh summ, Nauka. Gl. red.fiz.-mat. lit., M., 1987, 368 pp. | MR

[33] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric sums in number theory and analysis, De Gruyter expositions in mathematics, 39, Walter de Gruyter, Berlin–New York, 2004, 554 pp. | MR | Zbl

[34] Wang Yuan, Diophantine Equations and Inequalities in Algebraic Number Fields, Springer Verlag, Berlin–Heidelberg, 1991, 168 pp. | MR | Zbl

[35] Mardzhanishvili K. K., “Ivan Matveevich Vinogradov (k vosmidesyatiletiyu so dnya rozhdeniya)”, Usp. mat. nauk, 26:6 (1971), 3–6 | MR | Zbl

[36] Cassels J. W. S., Vaughan R. C., “Ivan Matveevich Vinogradov (obituary)”, Bull. London Math. Soc., 17 (1985), 584–600 | DOI | MR | Zbl

[37] Shafarevich I. R., “Patriarkh otechestvennoi matematiki”, Vestnik AN SSSR, 1991, no. 9, 96–100 | MR