Linear sums and the Gaussian multiplication theorem

O. V. Kolpakova ; V. N. Chubarikov

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 130-139.

Résumé

Estimations of linear sums with Bernoulli polynomial of the first degree are given. If the coefficient of the linear function is a irrational number with the bounded partial quotients, the arithmetical sum has the “squaring” estimation. The Roth's theorem gives the similar estimation for all algebraic number, but the constants in estimations be nonefficient. New difficulties appears for sums over primes. Their are connected with the consideration of bilinear forms. Bibliography: 24 titles.

Détail

Keywords: arithmetical sums, the Gaussian multiplication theorem for the Euler's Gamma-function, the functional theorem of the Gaussian type, the Bernoulli polynomials, algebraic numbers, arithmetical sums over primes, the Roth's theorem.

Bibliographie

[1] Vinogradov I. M., Fundamentals of Number Theory, Nauka, M., 1981, 176 pp. | MR

[2] Vinogradov I. M., Method of trigonometric sums in Number Theory, Nauka, M., 1980, 144 pp. (in Russian) | MR

[3] Cassels J. W. S., Introduction to the theory of diophantine approximation, Ed. foreign of literature, M., 1961, 213 pp. | MR

[4] Roth K. F., “Rational approximations of algebraic numbers”, Mathematika, 2 (1955), 1–20 ; corrigendum, 168 | DOI | MR | Zbl

[5] Chubarikov V. N., “About one multiple trigonometric integrals”, Rep. USSR AS, 227:6 (1976), 1308–1310 | MR | Zbl

[6] Chubarikov V. N., “Multiple rational trigonometric sums and multiple integrals”, Mat. notes, 20:1 (1976), 61–68 | MR | Zbl

[7] Arkhipov G. I., Selected works, Publishing House Oryol State University, Orel, 2013, 464 pp.

[8] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., Theory of multiple trigonometric sums, Nauka, M., 1987, 368 pp. | MR

[9] Arkhipov G. I., Sadovnichy V. A., Chubarikov V. N., Lectures on mathematical analysis, Drofa, M., 2004, 640 pp.

[10] Vinogradov I. M., “A new method of estimation of trigonometrical sums”, Math. USSR-Sb., 43:1 (1936), 175–188

[11] Hua L.-K., “An improvement of Vinogradov's mean-value theorem and several applications”, Quart. J. Math., 20 (1949), 48–61 | DOI | MR | Zbl

[12] Arkhipov G. I., “A theorem on the mean value of the modulus of a multiple trigonometric sum”, Math. Notes, 17 (1975), 84–90 | DOI | MR | Zbl

[13] Arkhipov G. I., Chubarikov V. N., “Multiple trigonometric sums”, Izv. Akad. Nauk SSSR, Ser. Mat., 40:1 (1976), 209–220 | MR | Zbl

[14] Hua L.-K., “On an exponential sums”, J. Chinese Math. Soc., 2 (1940), 301–312 | MR | Zbl

[15] Chen J.-R., “On Professor Hua's estimate on exponential sums”, Acta Sci. Sinica, 20:6 (1977), 711–719 | MR | Zbl

[16] Romanov N. P., Number Theory and functional analysis, proceedings, ed. V. N. Chubarikov, Izd-vo Tom. un-ta, Tomsk, 2013, 478 pp.

[17] Shihsadilov M. Sh., “A class of oscillating integrals”, Vestnik Moskow Univ. Ser. mat., meh., 2015, no. 5, 61–63 | MR

[18] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., “Trigonometric integrals”, Izv. Akad. Nauk SSSR, Ser. Mat., 43:5 (1979), 971–1003 | MR

[19] Titchmarsh E. C., The Theory of the Riemann Zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986 | MR

[20] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric Sums in Number Theory and Analysis, De Gruyter expositions in mathematics, 39, Berlin–New York, 2004 | MR | Zbl

[21] Hua L.-K., “Additive theory of prime numbers”, Trudy MIAN SSSR, 22, 1947, 1–179 | MR

[22] Montgomery H. L., Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysys, CBMS, Regional Conference Series in Mathematics, 84, 1994 | DOI | MR

[23] Hua L.-K., “On the number of solutions of Tarry's problem”, Acta Sci. Sinica, 1 (1953), 1–76 | MR

[24] Chubarikov V. N., “Linear arithmetic sums and Gaussian multiplication theorem”, Azerbaijan-Turkey-Ukrainian Int. Conf. “Mathematical Analysis, Differential Equations and their Applications”, Abstracts (September 08–13, 2015, Baku, Azerbaijan), 2015, 38