THE ARITHMETIC SUM AND GAUSSIAN MULTIPLICATION THEOREM
https://doi.org/10.22405/2226-8383-2015-16-2-231-253
Abstract
The paper presents the fundamentals of the theory of arithmetic sums and oscillatory integrals of polynomials Bernoulli, an argument that is the real function of a certain differential properties. Drawing an analogy with the method of trigonometric sums I. M. Vinogradov. The introduction listed problems in number theory and mathematical analysis, which deal the study of the above mentioned sums and integrals. Research arithmetic sums essentially uses a functional equation type Gauss theorem for multiplication of the Euler gamma function. Estimations of the individual arithmetic the amounts found indicators of convergence of their averages. In particular, the problems are solved analogues Hua Loo-Keng for one-dimensional integrals and sums.
Keywords
arithmetic sum oscillatory integrals,
polynomials Bernoulli,
Gauss theorem for multiplication of the Euler gamma function,
functional equation,
the average values of the convergence exponent arithmetic sums and oscillatory integrals,
Vinogradov’s method of trigonometric sums, problems Hua Loo-Keng
Supporting agencies: This work was financially supported by RFBR, grant N HK-13-01-00835
About the Author
V. N. Chubarikov
Механико-математический факультет,
Московский государственный университет имени М. В. Ломоносова
Russian Federation
Russian Federation
References
1. Vinogradov, I. M. 1981, "Osnovy teorii chisel."(Russian) [Foundations of the theory of numbers] Ninth revised edition “Nauka”, Moscow 176 pp.
2. Vinogradov, I. M. 1980, "Metod trigonometricheskikh summ v teorii chisel." , (Russian) [The method of trigonometric sums in the theory of numbers] Second edition. “Nauka”, Moscow, 144 pp.
3. Chubarikov, V. N. 1976, "On a multiple trigonometric integral" , Dokl. Akad. Nauk SSSR, vol. 227, № 6, pp. 1308–1310.
4. Chubarikov, V. N. 1976, "Multiple rational trigonometric sums and multiple integrals" , Mat. Zametki , vol. 20, № 1 pp. 61–68.
5. Arkhipov, G. I. 2013, "Selected Works" , Orel: Publisher Oryol State University, 464 pp.
6. Arkhipov, G. I., Karatsuba, A. A. & Chubarikov, V. N. 1987, "Teoriya kratnykh trigonometricheskikh summ" , (Russian) [Theory of multiple trigonometric sums] “Nauka”, Moscow, 368 pp.
7. Arkhipov, G. I., Sadovnichii, V. A. & Chubarikov, V. N. 2006, "Lectures on mathematical analysis" , Drofa, Moscow, 640 pp.
8. Vinogradov, I. M. 1936, "A new method of estimation of trigonometrical sums" , Math. USSR-Sb., vol. 43, No. 1, pp. 175–188.
9. Hua, L.-K. 1949, "An improvement of Vinogradov’s mean-value theorem and several applications" , Quart. J. Math., vol. 20, pp. 48–61.
10. Arkhipov, G. I. 1975, "A theorem on the mean value of the modulus of a multiple trigonometric sum" , Math. Notes, vol. 17, pp. 84–90.
11. Arkhipov, G. I., Chubarikov, V. N. 1976, "Multiple trigonometric sums" , Izv. Akad. Nauk SSSR, Ser. Mat., vol. 40, No. 1, pp. 209–220.
12. Hua, L.-K. 1940, "On an exponential sums" , J. Chinese Math. Soc., vol. 2, pp. 301– 312.
13. Chen, J.-R. 1977, "On Professor Hua’s estimate on exponential sums" , Acta Sci. Sinica, vol. 20, No. 6, pp. 711–719.
14. Romanov, N. P. 2013, "Number Theory and Functional Analysis: Proceedings / Under total. Ed. V. N. Chubarikov." , Tomsk: Publishing house of Tomsk State University, 478 pp.
15. Shihsadilov, M. Sh. 2015, "A class of oscillating integrals" , Vestnik Moskov. Univ. Ser. Mat., Mech. № 5, pp. 61–63.
16. Arkhipov, G. I., Karatsuba, A. A. & Chubarikov, V. N. 1979, "Trigonometric integrals" , Izv. Akad. Nauk SSSR, Ser. Mat. vol. 43, No. 5, pp. 971–1003.
17. Titchmarsh, E. C. 1986, "The Theory of the Riemann Zeta-function" , 2nd ed., The Clarendon Press, Oxford University Press, New York.
18. Arkhipov, G. I., Chubarikov, V. N. & Karatsuba, A. A. 2004, "Trigonometric Sums in Number Theory and Analysis" , De Gruyter expositions in mathematics; 39, Berlin, New York.
19. Hua, L.-K. 1947, "Additive theory of prime numbers" , Trudy MIAN SSSR., vol. 22, pp. 1–179.
20. Montgomery, H. L. 1994, "Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysys" , CBMS, Regional Conference Series in Mathematics, No. 84.
21. Hua, L.-K. 1953, "On the number of solutions of Tarry’s problem" , Acta Sci. Sinica, vol. 1, pp. 1–76.
Review
For citations:
Chubarikov V.N. THE ARITHMETIC SUM AND GAUSSIAN MULTIPLICATION THEOREM. Chebyshevskii Sbornik. 2015;16(2):231-253. (In Russ.) https://doi.org/10.22405/2226-8383-2015-16-2-231-253
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