ANATOLY ALEKSEEVICH KARATSUBA
https://doi.org/10.22405/2226-8383-2015-16-1-32-51
Abstract
The authors set themselves two main objectives: to characterize the main stages of the life of the outstanding Russian mathematician, Honored Scientist of Russia, Professor of Moscow State University named after M. V. Lomonosov, head of the department of number theory Steklov Mathematical Institute, Doctor of Physical and Mathematical Sciences Anatolii Alexeevich Karatsuba and give a brief analysis of his scientific work, which has had a significant impact on the development of analytic number theory. Sufficiently detailed description of the research of Professor A. A. Karatsuba and his disciples on the analytic theory of numbers, which are allocated following the Karatsuba, three main areas: 1) trigonometric sums and trigonometric integrals; 2) the Riemann zeta function; 3) Dirichlet characters. A. A. Karatsuba, being a disciple of Professor N. M. Korobov, led the scientific schools and seminars on analytic number theory at Moscow State University named after M. V. Lomonosov. Among his many students defended their dissertations, with seven of them later became Doctor of Physical and Mathematical Sciences. Anatoly Alekseevich has published 158 scientific papers, including 4 monographs and a classic textbook on analytic number theory, was the translator of a number of fundamental scientific monographs. He was a member of the editorial board of the journal "Mathematical notes"and a member of the program committees of several international conferences on algebra and number theory.
References
1. Arkhipov, G. I. 1975, "A theorem on the mean value of the modulus of a multiple trigonometric sum" , Mat. Zametki, vol. 17, no. 1, pp. 143–153. (Russian)
2. Arkhipov, G. I. & Chubarikov, V. N. 1976, "Multiple trigonometric sums" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 40, no. 1, pp. 209–220, 223. (Russian)
3. Arhipov, G. I., Karacuba, A. A. & Chubarikov, V. N. 1979, "Exponent of convergence of the singular integral in the Terry problem" , Dokl. Akad. Nauk SSSR, vol. 248, no. 2, pp. 268–272. (Russian)
4. Arhipov, G. I., Karacuba, A. A. & Chubarikov, V. N. 1979, "Trigonometric integrals" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 43, no. 5, pp. 971–1003, 1197. (Russian)
5. Arkhipov, G. I. & Karatsuba, A. A. 1982 "On local representation of zero by a form" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 45, no. 5, pp. 948–961. (Russian)
6. Arkhipov, G. I. 1981, "The values of a singular series in a Hilbert-Kamke problem" , Dokl. Akad. Nauk SSSR, vol. 259, no. 2, pp. 265–267. (Russian)
7. Arkhipov, G. I., Karatsuba, A. A. & Chubarikov, V. N. 1987, "Teoriya kratnykh trigonometricheskikh summ" , [Theory of multiple trigonometric sums] Nauka, Moscow, 368 pp. . (Russian)
8. Arkhipov, G. I. & Chubarikov, V. N. 1997, "On the mathematical papers of Professor A. A. Karatsuba" , Proc. Steklov Inst. Math., vol. 218, pp. 1-–14.
9. Vinogradov, I. M. 1947, "The method of trigonometrical sums in the theory of numbers" , Trav. Inst. Math. Stekloff, vol. 23, 109 pp. (Russian)
10. Vinogradov, I. M. 1953, "Improvement of an estimate for the sum of the values χ(p + k)" , Izvestiya Akad. Nauk SSSR. Ser. Mat., vol. 17, pp. 285—290. (Russian)
11. Voronin, S. M. 1976, "The zeros of zeta-functions of quadratic forms" , Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov, vol. 142, pp. 135–147, 269. (Russian)
12. Voronin, S. M. 1977, "The zeros of zeta-functions of quadratic forms" , Dokl. Akad. Nauk SSSR, vol. 235, no. 2, pp. 257–258. (Russian)
13. Voronin, S. M. 1980, "Zeros of some Dirichlet series lying on the critical line" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 44, no. 1, pp. 63–91, 238. (Russian)
14. Karatsuba, A. A. 1960, "Solution to a problem in the theory of finite automatons" , Uspehi Mat. Nauk, vol. 15, no. 3 (93), pp. 157–159. (Russian)
15. Karatsuba, A. A. & Ofman, Yu. P. 1962, "Multiplication of Many-Digital Numbers by Automatic Computers" , Dokl. Akad. Nauk SSSR, vol. 145, № 2, pp. 293–294. (Russian)
16. Karatsuba, A. A. & Korobov, N. M. 1963, "A mean-value theorem" , Dokl. Akad. Nauk SSSR, vol. 149, № 2, pp. 245–248. (Russian)
17. Karatsuba, A. A. 1968, "Sums of characters, and primitive roots, in finite fields" , Dokl. Akad. Nauk SSSR, vol. 180, № 6, pp. 1287–1289. (Russian)
18. Karatsuba, A. A. 1970, "Estimates of character sums" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 34, № 1, pp. 20–30. (Russian)
19. Karatsuba, A. A. 1970, "Sums of characters with prime numbers" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 34, № 2, pp. 299–321. (Russian)
20. Karatsuba, A. A. 1970, "The distribution of products of shifted prime numbers in arithmetic progressions" , Dokl. Akad. Nauk SSSR, vol. 192, № 4, pp. 724–727. (Russian)
21. Karatsuba, A. A. 1971, "Distribution of power residues and non-residues in additive squences" , Dokl. Akad. Nauk SSSR, vol. 196, № 4, pp. 759–760. (Russian)
22. Karatsuba, A. A. 1972, "A uniform estimate for the remainder term in Dirichlet’s problem of divisors" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 36, № 3, pp. 475–483. (Russian)
23. Karatsuba, A. A. 1973, "Mean value of the modulus of a trigonometric sum" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 37, № 6, pp. 1203–1227. (Russian)
24. Karatsuba, A. A. 1973, "Lower bounds for sums of the characters of polynomials" , Mat. Zametki, vol. 14, № 1, pp. 67–72. (Russian)
25. Karatsuba, A. A. 1978, "Distribution of the values of Legendre symbols in polynomials with prime numbers" , Dokl. Akad. Nauk SSSR, vol. 238, no. 3, pp. 524–526. (Russian)
26. Karatsuba, A. A. 1978, "Sums of Legendre symbols of quadratic polynomials with prime numbers" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 42, no. 2, pp. 315—324, 470. (Russian)
27. Karatsuba, A. A. 1981, "On the distance between consecutive zeros of the Riemann zeta function that lie on the critical line" , Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov, vol. 157, pp. 49–63, 235. (Russian)
28. Karatsuba, A. A. 1983, "Osnovy analiticheskoi teorii chisel" , [Principles of analytic number theory] Second edition. Nauka, Moscow, 240 pp. (Russian)
29. Karatsuba, A. A. 1984, "Zeros of the function ζ(s) on short intervals of the critical line" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 48, no. 3, pp. 569–584. (Russian)
30. Karatsuba, A. A. 1984, "Distribution of zeros of the function ζ(1/2 + it)" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 48, no. 6, pp. 1214–1224. (Russian)
31. Karatsuba, A. A. 1985, "The function G(n) in Waring’s problem" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 49, no. 5, pp. 935–947, 1119. (Russian)
32. Karatsuba, A. A. 1985, "Zeros of the Riemann zeta function on the critical line" , Current problems in mathematics. Mathematical analysis, algebra, topology. Trudy Mat. Inst. Steklov, vol. 167, pp. 167–178, 277. (Russian)
33. Karatsuba, A. A. 1985, "Real zeros of the function ζ(1/2 + it)" , Uspekhi Mat. Nauk vol. 40, no. 4(244), pp. 171–172. (Russian)
34. Karatsuba, A. A. 1990, "On zeros of the Davenport-Heilbronn function lying on the critical line" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 54, no. 2, pp. 303–315. (Russian)
35. Karatsuba A. A. 1992 "On Zeros of the Davenport – Heilbronn function" , Proc. Amalfi Conf. Analytic Number Theory, pp. 271–293.
36. Karatsuba, A. A. 1992, "On the number of zeros of the Riemann zeta function lying on almost all short intervals of the critical line" , Izv. Ross. Akad. Nauk Ser. Mat., vol. 56, no. 2, pp. 372–397. (Russian)
37. Karatsuba, A. A. 1993, "On zeros of arithmetic Dirichlet series without an Euler product" , Izv. Ross. Akad. Nauk Ser. Mat., vol. 57, no. 5, pp. 3–14. (Russian)
38. Karatsuba, A. A. 1995, "Analogues of Kloosterman sums" , Izv. Ross. Akad. Nauk Ser. Mat., vol. 59, no. 5, pp. 93–102. (Russian)
39. Karatsuba A. A. 1995, "Complex analysis in number theory" , Boca Raton: CRC Press, 187 pp.
40. Karatsuba, A. A. 1996, "A density theorem and the behavior of the argument of the Riemann zeta function" , Mat. Zametki, vol. 60, no. 3, pp. 448–449. (Russian)
41. Karatsuba, A. A. 1996, "On the function S(t)" , Izv. Ross. Akad. Nauk Ser. Mat., vol. 60, no. 5, pp. 27–56. (Russian)
42. Karatsuba, A. A. 1997, "Analogues of incomplete Kloosterman sums and their applications" , Tatra Mountains Math. Publ., vol. 11, pp. 89–120.
43. Karatsuba, A. A. 1999, "Double Kloosterman sums" , Mat. Zametki, vol. 66, no. 5, pp. 682–687. (Russian)
44. Karatsuba, A. A. 2000, "Sums of characters with prime numbers and their applications" , Tatra Mountains Math. Publ., vol. 20, pp. 155–162.
45. Karatsuba, A. A. 2000, "Character sums with weights" , Izv. Ross. Akad. Nauk Ser. Mat., vol. 64, no. 2, pp. 29–42. (Russian)
46. Karatsuba, A. A. 2001, "Lower bounds for the maximum modulus of ζ(s) in small domains of the critical strip" , Mat. Zametki, vol. 70, no. 5, pp. 796–797. (Russian)
47. Karatsuba, A. A. 2004, "On lower bounds for the maximum modulus of the Riemann zeta function on short intervals of the critical line" , Izv. Ross. Akad. Nauk Ser. Mat., vol. 68, no. 6, pp. 99–104. (Russian)
48. Karatsuba, A. A. & Karatsuba, E. A. 2009, "Application of ATS in a quantumoptical model" , Analysis and Mathematical Physics: Trends in Mathematics, pp. 211–232.
49. Karatsuba, A. A. & Karatsuba, E. A. 2009, "Resummation formula for collapse and revival in Jaynes – Cummings model" , J. Phys. A: Math. Theor., vol. 42, 195304, 16.
50. Mozer, Jan 1976, "A certain Hardy-Littlewood theorem in the theory of the Riemann zeta-function" , Acta Arith., vol. 31, no. 1, pp. 45–51. (Russian)
51. Mozer, Yan 1983, "Sharpening of the Hardy-Littlewood theorem on the density of zeros of the function ζ(1/2 + it)" , Acta Math. Univ. Comenian, vol. 42/43, pp. 41—50. (Russian)
52. Mozer, Ya. 1984, "Some corollaries of the Riemann-Siegel formula" , International conference on analytical methods in number theory and analysis (Moscow, 1981). Trudy Mat. Inst. Steklov, vol. 163, pp. 183–186. (Russian)
53. Davenport H. & Heilbronn H. 1936, "On Zeros of certain Dirichlet series" , J. Lond. Math. Soc., vol. 11, pp. 181–185 and pp. 307–312.
54. Davenport H. & Lewis D. J. 1963, "Primitive roots in finite fields" , Rend. Circolo Mat. di Palermo, vol. 12, pp. 129–136.
55. Hardy G. H. & Littlewood J. E. 1918, "Contributions to the theory of Riemann zeta-function and the theory of distribution of primes" , Acta Math., vol. 41, pp. 119–196.
56. Hardy G. H. & Littlewood J. E. 1921, "The zeros of Riemann’s zeta-function on the critical line" , Math. Zs., vol. 10, pp. 283–317.
57. Richert H. E. 1960, "Einf¨uhrung in die Theorie der Starken Rieszschen Summierbarkeit von Dirichletreihen" , Nachr. Akad. Wiss. G¨ottingen (Math.- Physik), pp. 17–75.
58. Selberg A. 1942, "On the zeros of Riemann’s zeta-function" , Skrifter utgitt av Det Norske Videnskaps–Akademi i Oslo. I. Mat.-Naturv. Klasse, vol. 10, pp.
59.
Review
For citations:
Arkhipov G.I., Chubarikov V.N. ANATOLY ALEKSEEVICH KARATSUBA. Chebyshevskii Sbornik. 2015;16(1):32-51. (In Russ.) https://doi.org/10.22405/2226-8383-2015-16-1-32-51
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