Skip to main content
Log in

I.M. Vinogradov’s method in number theory and its current development

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The paper presents a survey of the main results of I.M. Vinogradov.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. I. Arkhipov, “Investigations on the Hilbert–Kamke problem,” Chebyshev. Sb. 7 (1), 65–143 (2006).

    MathSciNet  MATH  Google Scholar 

  2. G. I. Arhipov and A. A. Karatsuba, “On local representation of zero by a form,” Izv. Akad. Nauk SSSR, Ser. Mat. 45 (5), 948–961 (1981) [Math. USSR, Izv. 19 (2), 231–240 (1982)].

    MathSciNet  MATH  Google Scholar 

  3. G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, Theory of Multiple Trigonometric Sums (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  4. J. W. S. Cassels and R. C. Vaughan, “Obituary: Ivan Matveevich Vinogradov,” Bull. London Math. Soc. 17 (6), 584–600 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Chamizo and H. Iwaniec, “On the sphere problem,” Rev. Mat. Iberoam. 11 (2), 417–429 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. N. Chubarikov, “Estimates of multiple trigonometric sums with prime numbers,” Izv. Akad. Nauk SSSR, Ser. Mat. 49 (5), 1031–1067 (1985) [Math. USSR, Izv. 27 (2), 323–357 (1986)].

    MathSciNet  MATH  Google Scholar 

  7. V. N. Chubarikov, “The arithmetic sum and Gaussian multiplication theorem,” Chebyshev. Sb. 16 (2), 231–253 (2015).

    MathSciNet  Google Scholar 

  8. D. R. Heath-Brown, “Lattice points in the sphere,” in Number Theory in Progress: Proc. Int. Conf., Zakopane (Poland), 1997, Vol. 2: Elementary and Analytic Number Theory (W. de Gruyter, Berlin, 1999), pp. 883–892.

    Google Scholar 

  9. E. Landau, Vorlesungen über Zahlentheorie (S. Hirzel, Leipzig, 1927), Vol. 1.

    MATH  Google Scholar 

  10. D. A. Mit’kin, “An estimate for the number of terms in the Hilbert–Kamke problem,” Mat. Sb. 129 (4), 549–577 (1986) [Math. USSR, Sb. 57 (2), 561–590 (1987)].

    MathSciNet  MATH  Google Scholar 

  11. I. R. Shafarevich, “Patriarch of Russian mathematics,” Vestn. Akad. Nauk SSSR, No. 9, 96–100 (1991).

    Google Scholar 

  12. G. Szegö, “Beiträge zur Theorie der Laguerreschen Polynome. II: Zahlentheoretische Anwendungen,” Math. Z. 25, 388–404 (1926).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Vinogradov, “Gaussian sums and their application to the proof of the reciprocity law of quadratic residues,” Graduation Thesis (St. Petersburg. Imper. Univ., St. Petersburg, 1914), http://www.mi.ras.ru/persons/directors/vinogradov/Vinogradov.pdf

    Google Scholar 

  14. I. M. Vinogradov, “A new method for obtaining asymptotic expressions for arithmetic functions,” Izv. Akad. Nauk 11 (16), 1347–1378 (1917).

    Google Scholar 

  15. I. M. Vinogradov, “On the mean value of the number of classes of purely root forms of negative determinant,” Soobshch. Kharkov. Mat. Obshch., Ser. 2, 16 (1–2), 10–38 (1918).

    Google Scholar 

  16. I. M. Vinogradov, “Sur la distribution des résidus et des non-résidus des puissances,” Zh. Fiz.-Mat. Obshch. Perm. Univ. 1, 94–98 (1918).

    Google Scholar 

  17. J. M. Winogradow, “Sur un théorème général de Waring,” Mat. Sb. 31 (3–4), 490–507 (1924).

    Google Scholar 

  18. I. M. Vinogradov, “An elementary proof of a general theorem in analytic number theory,” Izv. Ross. Akad. Nauk 19 (16–17), 785–796 (1925).

    Google Scholar 

  19. I. M. Vinogradov, “On the distribution of indices,” Dokl. Akad. Nauk SSSR A, No. 4, 73–76 (1926).

    Google Scholar 

  20. I. M. Vinogradov, “On the bound of the least non-residues of n-th powers,” Trans. Am. Math. Soc. 29 (1), 218–226 (1927).

    MathSciNet  MATH  Google Scholar 

  21. I. M. Vinogradov, “On Waring’s theorem,” Izv. Akad. Nauk SSSR, Otd. Fiz.-Mat. Nauk, No. 4, 393–400 (1928).

    Google Scholar 

  22. I. M. Vinogradov, “A new solution of Waring’s problem,” Dokl. Akad. Nauk SSSR 2 (6), 337–341 (1934).

    MATH  Google Scholar 

  23. I. M. Vinogradov, “On some new problems of the theory of numbers,” Dokl. Akad. Nauk SSSR 3 (1), 1–6 (1934).

    MathSciNet  Google Scholar 

  24. I. M. Vinogradov, “A new evaluation of G(n) in Waring’s problem,” Dokl. Akad. Nauk SSSR 4 (5–6), 249–253 (1934).

    MATH  Google Scholar 

  25. I. M. Vinogradov, “On the upper bound of G(n) inWaring’s problem,” Izv. Akad. Nauk SSSR, Otd. Mat. Estestv. Nauk, No. 10, 1455–1469 (1934).

    Google Scholar 

  26. I. M. Vinogradov, “New estimates for Weyl sums,” Dokl. Akad. Nauk SSSR 3 (5), 195–198 (1935).

    MATH  Google Scholar 

  27. I. M. Vinogradov, “Representation of an odd number as a sum of three primes,” Dokl. Akad. Nauk SSSR 15 (6–7), 291–294 (1937).

    Google Scholar 

  28. I. M. Vinogradov, “On the estimations of some simplest trigonometrical sums involving prime numbers,” Izv. Akad. Nauk SSSR, Ser. Mat. 3 (4), 371–398 (1939).

    MathSciNet  Google Scholar 

  29. I. M. Vinogradov, “An improvement of the estimation of sums with primes,” Izv. Akad. Nauk SSSR, Ser. Mat. 7 (1), 17–34 (1943).

    MathSciNet  Google Scholar 

  30. I. M. Vinogradov, “Improvement of an estimate for the sum of the values χ(p + k),” Izv. Akad. Nauk SSSR, Ser. Mat. 17 (4), 285–290 (1953).

    MathSciNet  Google Scholar 

  31. I. M. Vinogradov, “A new estimate for the function χ(1 + it),” Izv. Akad. Nauk SSSR, Ser. Mat. 22 (2), 161–164 (1958).

    MathSciNet  Google Scholar 

  32. I. M. Vinogradov, “On the number of integer points in a sphere,” Izv. Akad. Nauk SSSR, Ser. Mat. 27 (5), 957–968 (1963).

    MathSciNet  MATH  Google Scholar 

  33. I. M. Vinogradov, Special Variants of the Method of Trigonometric Sums (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  34. I. M. Vinogradov, The Method of Trigonometric Sums in Number Theory (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  35. I. M. Vinogradov and A. A. Karatsuba, “The method of trigonometric sums in number theory,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 168, 4–30 (1984) [Proc. Steklov Inst. Math. 168, 3–30 (1986)].

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. P. Mineev.

Additional information

Original Russian Text © M.P. Mineev, V.N. Chubarikov, 2017, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 296, pp. 7–23.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mineev, M.P., Chubarikov, V.N. I.M. Vinogradov’s method in number theory and its current development. Proc. Steklov Inst. Math. 296, 1–17 (2017). https://doi.org/10.1134/S0081543817010011

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543817010011

Navigation