A. V. Malyshev, Scientist And Teacher

Journal of Mathematical Sciences, Vol. 83, No. 5, 1997

A. V. MALYSHEV

S C I E N T I S T A N D TEACHER,

Aleksandr Vasil'evich Malyshev passed away on May 10, 1993. He was a top-level expert in number theory, the leading scientific worker in the St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences (POMI), Professor in the Department of High Algebra and Number Theory of the St. Petersburg State University (SPbGU), Doctor of Physics and Mathematics, and the originator and first editor of "Zapiski Nauchnykh Seminarov POMI." A. V. Malyshev was born on November 17, 1928 in St. Petersburg. In 1951, he was graduated with honors from the Department of Mathematics and Mechanics of the St. Petersburg State University. His graduation thesis (published in "Uspekhi Matematicheskikh Nauk") was carried out under the supervision of Professor B. A. Venkov, who was his first teacher in number theory and whom he always kept in remembrance. From that year, he was a scientific worker in POMI, where he worked for his whole life. He defended the thesis for the Candidate's degree in 1954 and the thesis for the Doctoral degree in 1961. Academician Yu. V. Linnik was a scientific adviser of Malyshev in POMI, and Aleksandr Vasil'evich was worked successfully under his direction. After the death of Linnik in 1972, Aleksandr Vasil'evich expended much energy and time in developing and propagating the ideas of Linnik. In addition to his scientific work, which will be surveyed below, Malyshev was known for his teaching and organizational activities. For many years he was a University Lecturer; he organized various special courses and seminars and took charge of the Seminar on number theory that was held at POMI for many years. At this seminar many theses of the degrees of Candidate and Doctor were approved. Malyshev had more than 20 pupils, almost all of whom successfully defended their theses. A. V. Malyshev was a member of scientific councils of POMI and SPbGU, of the Library of Academy of Sciences, of special councils on defense of theses for the degrees of Doctor and Candidate and acted as Opponent of many of those theses. He was a member of coordination committees of many All-Union Conferences and took part in organizing the Fourth All-Union Mathematical Congress in 1961 and the International Congress of Mathematics in Moscow in 1966. He attached special significance to editorial and publishing work. He was the initiator of publishing the series "Zapiski nauchnykh seminarov POMI" and the editor of the first volume of that series. Nine volumes of "Investigations on number theory" were published in "Zapiski" under his direction. For many years. Malyshev was the editor of the section on number theory of the Russian Mathematical Reviews; he was the editor of more than 30 books on topical problems in number theory, among them the selected papers of his teachers B. A. Venkov and Yu. V. Linnik; much attention and much effort was givert to this work. A. V. Malyshev was one of the leading authorities on various challenging problems in number theory, but he mainly focused on the theory of quadratic forms and the geometry of numbers. In the theory of quadratic forms his main concern was with the problem on integer representations of numbers by quadratic forms, i.e., the problem of studying the solutions of Diophantine equations of the form

where Q is a quadratic form in s variables, n is an integer, and also with generalizations of this problem

and other problems related to it. Of special interest to him was the problem on the existence of such representations and the distribution of them with respect to a given modulus and on the associated surface of second order, or, more precisely, the problem of finding an asymptotic formula (as n .~ oo) for the number of representations lying in a given residue class and in a given domain on the surface Q(x) = n. The complexity of this problem heavily depends on the number of variables s of the quadratic form under consideration. Thus, for s > 5, asymptotic formulas for the number of representations were obtained of Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 7-13. Original article submitted .

1072-3374/97/8305-0565 $18.00 9 1997 Plenum Publishing Corporation

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immediate application of the standard tools of analytic number theory: with the help of the method of circles and the method of trigonometric sums originated by Hardy, Littlewood, Ramanujan, and I. M. Vinogradov. For his part, Malt shev was able to refine significantly the technique of these methods applied to the problem under consideration, first of all, by doing more precise work with generalized Gauss sums that arise under this approach, which allowed him to use the best possible estimates of these sums obtained by A. Weil at that time. For positive quadratic forms this enabled him to obtain for the first time workable asymptot,.'c formulas for the number of representations by the forms in s = 4 variables; for s _> 5, this also enabled him to improve substantially the remainder term in the asymptotic formulas. Moreover, the formulas obtained also made it possible to solve the problem concerning the distribution of the representations with respect to a given modulus and over the surface of the corresponding many-dimensional ellipsoid. For indefinite quadratic forms Q in s > 4 variables, the same methods allowed Aleksandr Vasil'evich to obtain an asymptotic formula for the number of integral points in a domain on the second-order surface Q ( x ) = n. Moreover, Malyshev derived formulas that enabled him to study and evaluate efficiently the principal term of these asymptotic expansions. However, the remaining most difficult case of s - 3, i.e., the problem of representation of the numbers by ternary quadratic forms, commanded the major attention and efforts of A. V. Malyshev (the case s = 2 is distinguished for two reasons: first, in this case analytic methods are practically inapplicable because we have no possibility of assuming that the distribution of the representations of the numbers by binary forms is uniform; on the other hand, the whole theory is quite different because of the existence of the composition of binary forms). In order to solve this problem, for many years Aleksandr Vasil'evich and his pupils developed the so-called discrete ergodic method. The foundation of this method was laid by Academician Yu. V. Linnik, who originated it to study the distribution of integral points on the sphere x 2 + y2 + z = n and on the hyperboloid x y - z 2 = n. The basis for this method (in the form in which we consider it now) is the examination of the refined structure of the action of rational automorphisms of the form Q on integer representations of the number n by the form Q. As is well known, the proper automorphisms of the form Q correspond to elements of the even Clifford algebra of the form Q. For ternary forms, the even Clifford algebra is the algebra of generalized quaternions; the study of arithmetic properties of the orders 53 in this algebra enables one to introduce an operator T acting on the set of primitive representations of the number n by the genus of the form Q'. The action of this operator is described in terms of quaternions X from the order 53 with given norm N ( X ) k. The norm N in the quaternion algebra is a quadratic function defined on a four-dimensional space. Therefore, the description of quaternions of norm k in the order 53 reduces to the problem on representation of the number k by a quaternary quadratic form. As discussed above, Malyshev's results for s -. 4, together with a very fine estimate from above for the number of primitive representations Q ( x ) = n that possess the additional property of divisibility by a fixed quaternion X of norm k in the order ~ (the so-called "key lemma" of the method), enabled him to establish that the operator T has the following "ergodlc " " property,9 for " almost all " pnmatlve . . representatxons . . Q ( x ) = n, the vectors x, T x , T 2 x , . . . , T " x , . . . distribute asymptotically uniformly with respect to the genus classes of the form Q (more precisely, with respect to the spinor genus classes, which is related to the specific character of the case s = 3), with respect to the residue classes of a given modulus and on the surface Q(x) = n (with respect to an invariant measure). Hence, it is easy to derive that the primitive representations themselves also distribute asymptotically uniformly, which makes it possible to obtain an asymptotic formula for the number of representations of the number n by a ternary quadratic form Q that lie in a given residue class and in a given domain on the surface Q ( x ) = n. Unfortunately, under this approach a somewhat weak lowering in the remainder term occurs, so that this formula requires a number of additional assumptions (of the type of a weakened version of the extended Pdemann hypothesis). Nevertheless, until very recently this method was the only method that allowed for deriving an unconditional asymptotic formula, if only for some ternary quadratic forms. (Recently these results were radically improved because of the fundamental paper of H. Iwaniec on estimation of the Fourier coefficients of modular cusp forms of half-integral weight and because of the papers of some other authors that develop this approach, which leads, in particular, to a lowering in the degree of the remainder term of the asymptotic formula considered above.) 566

Originally, Aleksandr Vasil'evich Malyshev applied the discrete ergodic method to study the positive ternary forms, and with the help of this method he obtained an estimate precise (up to order of magnitude) from below for the number of primitive representations of the numbers by such forms. To this end, he needed to develop considerably divisibility theory in the orders of the generalized quaternions used in his proofs. These papers, together with the above results for positive forms in s > 4 variables, lay the foundation of the Doctoral thesis of A. V. Malyshev. Later, Malyshev explored the discrete ergodic method further together with his pupils; some work along this line of inquiry was done by his pupils, but in such cases Aleksandr Vasil'evich also offered his guidance in improving the results obtained and in preparing them for publication. The most advanced results for positive ternary forms were obtained by A. V. Malyshev in collaboration with Yu. G. Teterin, and later this line of investigation was continued by Teterin independently. The study of the representations of n by indefinite forms heavily depends on whether the surface Q ( z ) = n is a hyperboloid of one or two sheets. For a hyperboloid of two sheets the main results were obtained by Malyshev in collaboration with U. M. Pachev; later on U. M. Pachev also continued these studies independently. The considerably more complicated case of a hyperboloid of one sheet was primarily examined for the simplest equation xl z3 - x 2 = n, n < 0, by another pupil of Academician Yu. V. Linnik, Boris Faddeevich Skubenko, who also passed away recently. Later on, Ma[yshev, in cooperation with his Vietnamese pupil Nguen Ngor Goi, continued these investigations, revising the results obtained, which led to an extension of the results to certain forms of a more general type. Other pupils of A. V. Malyshev also took part in developing the discrete ergodic method. In particular, N. N. Belova worked in this area of research independently and in collaboration with Malyshev. E. V. Podsypanin obtained a number of results on the distribution of integral points on surfaces of second order that are necessary in order to apply the method in the case of indefinite ternary forms. Moreover, investigations were pursued in order to realize Linnik's program of applying this method to study the ergodic properties of algebraic number fields; here, some interesting results were obtained as well. As noted above, the technique applied in the discrete ergodic method invited further development of divisibility theory in the orders of generalized quaternions. Moreover, these studies led to the statement of a series of problems that are of interest in themselves for the arithmetic of quaternions. In this area of research, Malyshev worked in collaboration with his pupil M. N. Kubenskii, who obtained a number of interesting results. Aleksandr Vasil'evich took an active interest in other approaches to the analytic theory of quadratic forms, in particular, to the theory of modular forms. Here, his prime interest was in the problem of estimating the Fourier coefficients of modular forms, which by applying it to the problem of representations of numbers (or forms) by quadratic forms, enabled him to obtain an asymptotic formula for the number of representations. The problems of expanding the theta series in special bases of the space of modular forms, closely related to the classical problems of the so-called "exact" formulas in the problem on representation of numbers by quadratic formulas, were also of primary concern to A. V. Malyshev. A. B. Voronetski was one of the pupils of Malyshev who actively developed the theory of modular forms and applied it to quadratic forms. In Izhevsk, Voronetski organized a group of researchers who were working in this area and fruitfully collaborating with Malyshev, who actually was their supervisor. Thus, V. V. Golovin and A. Merslyakov, members of this group, obtained a number of new results for Hilbert modular forms under the supervision of Aleksandr Vasil'evich. Investigations of "exact" formulas were pursued by many of the pupils of Malyshev (in collaboration with him). In particular, E. A. Kashina has obtained interesting results in this area. A. V. Malyshev's attention was also given to other problems of analytic number theory, including the study of systems of Diophantine equations and inequalities with large number of unknowns. His comprehensive survey (in collaboration with E. V. Podsypanin) in the series "Itogi Nauki i Tekhniki" was devoted to these problems. The geometry of numbers was another wide area of study of A. V. Malyshev. The most interesting results were related to the Minkowski conjecture on the critical determinant of the domain IXI v + IYI p _< 1 (p is a real parameter), which was completely proved by Malyshev (in cooperation with his pupils). To this end, Malyshev developed the techniques that enabled him to reduce the Minkowski conjecture to the problem of finding minimums of some simpler function and further to localize these minimums with the 567

help of estimates of partial derivatives in relevant domains. In a number of papers (in collaboration with A. V. Voronetskii and K. I. Grishmanovskaya, as well as with U. M. Pachev and A. Ch. Fidarova) in which a computer was actively used, Malyshev used this method systematically, successively narrowing the range of the parameter p for which the conjecture remained unproved. Later, M. M. Glasynov and A. C. Golovanov took part in this work. As a result, a complete proof of the Minkowski conjecture was obtained in 1986. Malyshev was very proud of this work. A. V. Malyshev took an active part in studies in other areas of analytic number theory and the geometry of numbers. In particular, of interest to him were the questions concerning an arithmetic minimum of the product of linear nonhomogeneous forms closely related to the so-called "nonhomogeneous Minkowski conjecture." These questions were studied by his colleague B. F. Skubenko, mentioned above. Here, Aleksandr Vasil'evich, together with his pupils I. V. II'in and A. K. Andriyasyan, succeeded in improving the results concerning estimates of the so-called "Chebotarev type" of this arithmetic minimum. A. V. Malyshev concentrated much attention to such an area of number theory as the geometry of quadratic forms that lies at the interfaces between the theory of quadratic forms and the geometry of numbers. Here, under his supervision, one of his first pupils, P. P. Tammela, obtained a number of final results concerning the reduction theory of quadratic forms, and his last pupil, E. B. Orlovskaya, treated the problem on the minimum of the Epstein zeta function of ternary quadratic forms. Colleagues, pupils, and friends of Aleksandr Vasil'evich will always keep his memory alive and will continue to develop his scientific ideas. Z. I. Borevich, A. P. Oskolkov, E. V. Podsypanin, A. I. Skopin, Yu. G. Teterin, A. V. Yakovlev. ALEKSANDR VASIL'EVICH MALYSHEV's PRINTED PAPERS* 1952 1. "On the Minkowski-Hlavka theorem concerning a star body," ~ p .

Mat. Nauk, 7, No. 2, 168-171

(1952). 2. "On the representation of large numbers by positive ternary quadratic forms," Dokl. Akad. Nauk SSSR, 87, No. 2, 175-178 (1952). 1953 3. "An asymptotic law for the representation of numbers by some positive ternary quadratic forms," Dokl. Akad. Nauk SSSR, 93, No. 5, 771-774 (1953). 4. "On the representation of numbers by positive ternary quadratic forms," Dokl. Akad. Nauk SSSR, 89, No. 3, 405-406 (1953). 5. "On integer points on a sphere," Dokl. Akad. Nauk SSSR, 89, No. 2, 209-211 (1953). 6. (In collaboration with Yu. V. Linnik) "Applications of the arithmetic of quaternions to the theory of ternary quadratic forms and to the decomposition of numbers into cubes," Usp. Mat. Nauk, 8, No. 5, 3-71 (1953); Corrections, ibid., 10, No. 1,243-244 (1955). 1954

7. On Integer Points on Ellipsoids [in Russian], Thesis, Leningr. Univ., Leningrad (1954). 8. "On integer points on ellipsoids," Usp. Mat. Nauk, 9, No. 3,253-255 (1954). 1956

"An asymptotic distribution of integer points on some ellipsoids," Proc. of the 3rd All-Union Math. Syrup., Vol. 1, Nauka, Moscow (1956), pp. 7-8. 10. "On integer points on ellipsoids," Ve~tn. LGU, Set. Mat., Mekh., Astron., No. 4, 18-34 (1956). 11

1957

11. "An asymptotic distribution of integer points on some ellipsoid," Izv. Akad. Nauk SSSR, Ser. Mat., 21, No. 4, 457-500 (1957); Letter to editorial staff, ibid., 22, No. 5, p. 735 (1958). The bibliography was compiled by E. G. Vinogradova and V. N. Chugueva.

568

12. "On the distribution of integer points on a four-dimensional sphere," Dokl. Akad. Nauk SSSR, 114, No. 1, 25-28 (1957). 1958

13. "On the representation of large numbers by positive ternary quadratic forms of odd coprime invariants," Dokl. Akad. Nauk SSSR, 118, No. 6, 1078-1080 (1958). 14. "On the relationship between the theory of distribution of L-series zeroes and the arithmetic of ternary quadratic forms," Dokl. Akad. Nauk SSSR, 122, No. 3, 343-345 (1958). 1959

15. "On the theory of ternary quadratic forms. 1. The arithmetic of hermitions," Vestn. L G U, Set. Mat., Mekh., Astron., 2, 55-71 (1959). 16. "On the theory of ternary quadratic forms. 2. On a theorem of Linnik," Vestn. LGU, Ser. Mat., Mekh., Astron., 3, 63-70 (1959). 17. "Leningrad Mathematical Society, " Usp. Mat. Nauk, 14, No. 6, 212 (1959). 18. "On the representation of integers by positive quadratic forms in four or more variables. 1~ ~ Zzv. Akad. Nauk SSSR, Ser. Mat., 23, No. 3, 337-364 (1959). 1960

19. "On the theory of ternary quadratic forms. 3. On the representation of large numbers by positive forms of odd coprime invariants," Vestn. Leningr. Univ. Mat., Mekh., Astron., 1, 70-84 (1960). 20. "On the theory of ternary quadratic forms. 4. On a connection with the Riemann hypothesis," Vestn. LGU, Set. Mat., Mekh., Astron., Vyp. 2, 14-27 (1960). 21. "On quadratic forms in an arbitrary field. A generalization of Pail's theorem," Usp. Mat. Nauk, 15, No. 3, 167-172 (1960). 22. "On quadratic forms over an arbitrary field," Usp. Mat. Nauk, 15, No. 6, 182-183 (1960). 23. "On the representation of integers by positive quadratic forms in four or more variables," Dokl. Akad. Nauk SSSR, 133, No. 6, 1294-1297 (1960). 24. "On Gauss and Kloosterman sums," Dokl. Akad. Nauk SSSR, 133, No. 5, 1017-1020 (1960). 25. "Generalized Kloosterman sums and their estimates, 9 " Vestn. L GU, Set. Mat., Mekh., Astron., 3, 5976 (1960); "On Fourier coefficients of modular forms (Remark to the paper "Generalized Kloostermann sums and their estimates"), 9 " Zap Nauchn. Semin. LOMI, 1,140-163 (1966). 1961

26. (In collaboration with D . K . Faddeev), " Boris Alekseevich Venkov (on the occasion of his sixtieth birthday). Bibliography," Usp. Mat. Nauk, 16, No. 4, 235-240 (1961). 27. "A new variant of the proof of the Stuff-Minkowski theorem on the finiteness of the number of edges of a Hermite reduction region," Usp. Mat. Nauk, 16, No. 2, 127-129 (1961). 28. On the Representation of Integers by Positive Quadratic Forms [in Russian], Doctoral Degree Thesis, Leningr. Univ., Leningrad (1961). 1962

29. "On the representation of integers by positive quadratic forms," Tr. Mat. Inst. Steklov, 65, 1-212 (1962). 1963

30. "On the distribution of integer points on surfaces of the second order," Ushen. Zap. KabardinoBalkarsk. Univ., Ser. Fiz.-Mat. Nauk, '19, 195-196 (1963). 1964 31. "On the representation of integers by indefinite quadratic forms," Volzhsk. Mat. Sb., 2 91-95 (1964).

32. "Representation of integers by quadratic forms," in: Proc. 4th All-Union Math. Congr. (Leningrad, 1961), Vol. 2, Nauka, Leningrad (1964), pp. 118-124. 33. (In collaboration with Yu. V. Linnik), "An elementary proof of the Kleerterman-Tartakovskii theorem on the representation of numbers by positive quaternary forms," in: Proc. 4th All- Union Math. Congr. (Leningrad, 1961) Vol. 2, Nauka, Leningrad (1964), pp. 116-117. 569

1965

34. (In collaboration with I. A. Ibragimov and V. V. Petrov), "Yurii Vladimirovich Linnik (on the occasion of his fiftieth birthday)," Usp. Mat. Nauk, 20, No. 2, 221-236 (1965). 1966

35. "On a weighted number of integer points lying on a surface of second order," Zap. Nauchn. Semin. LOMI, 1 6-83 (1966). 36. (In collaboration with B. Z. Moroz), "The distribution of integer points on surfaces of second order," in: Internat. Math. Congr. Section 3. Number Theory: Abstracts, Moscow (1966), p. 18. 1968

37. "Quadratic forms. Geometry of numbers," in: The History of Mathematics of Our Country, Vol. 3, Naukova Dumka, Kiev (1968), pp. 248-266. 38. (In collaboration with A. I. Vinogradov, B. V. Levin, N. P. Romanov, and N. G. Chudakov), "Mark Borisovich Barban (obituary)," Usp. Mat. Nauk, 24, No. 2, 213-216 (1969). 39. (In collaboration with A. I. Vinogradov, B. V. Levin, N. P. Romanov, and N. G. Chudakov), "Aleksandr Osipovich Gel'fand (1906-1968)," Sib. Mat. Zh., 10, No. 2, 478-479 (1969). 40. (In collaboration with E. V. Podsypanin), "Some unsolved problems of the geometry of numbers," in: All-Union Syrup. on Number Theory (Alma-Ata, 1969), Abstracts, Alma-Ata (1969), p. 49. 41. "Applications of computers in number theory," in: All-Union Syrup. on Number Theory (Alma-Ata, I969), Abstracts, Alma-Ata (1969), p. 48. 1971

42. (In collaboration with A. N. Andrianov et al.), "On N. I. Gavrilov's book 'The Riemann Problem on the Distribution of Roots of the Zeta-Function'," Usp. Mat. Nauk, 26, No. 3, 238-247 (1971). 1973 43. "A remark on a star set," Zap. Nauchn. Semin. LOMI, 33, 94-96 (1973). 44. (In collaboration with I. A. Ibragimov, S. M. Lozinski~, V. V. Petrov, Yu. V. Prokhorov, N. A. Sapogov, and D. K. Faddeev), "Yurii Vladimirovich Linnik (1915-1972) (obituary)," Usp. Mat. Nauk, 28, No. 2, 197-213 (1973). 45. "Mordell's method of reciprocal lattices in the geometry of numbers," Zap. Nauchn. Semin. LOMI, 33, 97-115 (1973). 1974 (In collaboration with E. V. Podsypanin), "Analytic methods in the theory of systems of Diophantine equations and inequalities with a large number of unknowns," Itogi Nauki Tekhniki, Algebra, Topol. Geom., 12, 5-50 (1974). 47. (In collaboration with A. B. Voronetskii), "A proof of Minkowski's conjecture concerning the critical determinant of the region Ixl p + lyl p < 1 for p > 6," in: Probler,,.~ of Analytic Number Theory and Its Applications. All-Union Conf. (Vil'nyus, 1974): Abstracts, Vil'nyus (1974), pp. 50-53. 48. "On formulas for the number of representations of numbers by positive quadratic forms (problems)," in: Actual Problems of Analytic Number Theory, Nauka i Tekhnika, Minsk (1974), pp. 119-137. 49. "The ergodic method of Yu. V. Linnik in number theory: Results, perspectives," in: Problems of Analytic Number Theory and Its Applications. All-Union Conf. (Vil'nyus, 197~) Abstracts, Vil'nyus (1974), pp. 168-169.

46.

1975

50. "A new version of the ergodic method of Yu. V. Linnik in number theory," Zap. Nauchn. Semin. LOMI, 50, 179-186 (1975). 51. (In collaboration with V. E. Voskresenskii and G. I. Perel'muter), "Nikolai Grigor'evich Chudakov (on his seventieth birthday)," Usp. Mat. Nauk, 30, No. 3, 195-197 (1975). 52. (In collaboration with T. G. Fursa), "Bibliography of scientific works of Yu. V. Linnik," Acts Arith., 27, 11-35 (1975). 570

53. (In collaboration with D. K. Faddeev and S. M. Lozinskii), "Yurii V. Linnik (1915-1972). A biographical note," Acta Arith., 27, 1-2 (1975). 54. "Yu. V. Linnik's ergodic method in number theory," Acta Arith., 27, 555-598 (1975). 55. "Yu. Linnik's works in number theory," Acta Arith., 27, 3-10 (1975). 56. (In collaboration with A. B. Voronetskii), "The proof of Minkowski's conjecture concerning the critical determinant of the region Ixl p + ly[P < 1 for p > 6," Acta Arith., 27, 447-458 (1975). 1976 57. (In collaboration with A. B. Voronetskii), "On a simultaneous representation of a pair of numbers by sums of integers and their squares," Tr. Mat. Inst. Steklov, 142, 122-134 (1976). 1977

58. "Binary quadratic form," in: Encyclopediya o.f Mathematics, Vol. 1, Sovet. Entsiklopedia, Moscow (1977), pp. 486-487. 59. "Binary form," in: Encyclopediya o/Mathematics, Vol. 1, Sovet. Entsiklopedia, Moscow (1977), pp. 487-488. 60. "Geometry of numbers," in: Encyclopediya of Mathematics, Vol. 1, Sovet. Entsiklopedia, Moscow (1977), pp. 944-950. 61. (In collaboration with K. I. Grishmanovskaya, U. M. Pachev, and A. Ch. Fidarova), "A proof of Minkowski's conjecture concerning the critical determinant of the region Ixip + lyl p < 1 for 5 < p < 6," Zap. Nauchn. Semin. LOMI, 67, 95-107 (1977). 62. "On the application of a computer to a proof of Minkowski's conjecture in the geometry of numbers," Zap. Nauchn. Semin. LOMI, 71, 163-180 (1977); "On the critical determinant of the region Ix]p + lyl p < 1 (Supplement to the article "On the application of a computer to a proof of Minkowski's conjecture in the geometry of numbers")," Zap. Nauchn. Semin. LOMI, 82, 29-32 (1979). 63. (In collaboration with A. B. Voronetskii and K. I. Grishmanovskaya), "An application of a computer to a proof of Minkowski's conjecture concerning the critical determinant of the region Izl p + lyl p < 1," in: All-Union School on Number Theory: Abstracts, Dushambe (1977), p. 26. 64. "The Markov and the Lagrange spectrum (a survey of the literature)," Zap. Nauchn. Semin. LOMI, 67, 5-38 (1977). 1978 65. "On the application of a computer to a proof of Minkowski's conjecture," in: Artificial Intelligence and Automation of Investigations m Mathematics. All-Union Syrup. (Kiev, 1978), Abstracts, Kiev (1978), pp. 59-60. 1979

66. "The discrete ergodic method of Yu. V. Linnik and its subsequent development," in: Selected Works. Number Theory. Ergodic Method and L-Functions, Nauka, Leningrad (1979), pp. 418-430. 67. "Star body," in: Encyclopediya of Mathematics, Vol. 2, Sovet. Entsiklopedia, Moscow (1979), pp. 446447. 68. "Quadratic forms," in: Encyclopediya of Mathematics, Vol. 2, Sovet. Entsiklopedia, Moscow (1979), pp. 776-782. 69. "Reduction of quadratic forms," in: Encyclopediya of Mathematics, Vol. 2, Sovet. Entsiklopedia, Moscow (1979), pp. 788-791. 70. "Quaternary quadratic forms," in: Encyclopediya of Mathematics, Vol. 2, Sovet. Entsiklopedia, Moscow (1979), p. 837. 71. (In collaboration with U. M. Pachev), "On the representation of integers by positive ternary quadratic forms (a new modification of the discrete ergodic method), " Zap. Nauchn. Semin. L OMI, 82, 33-37 (1979). 72. (In collaboration with U. M. Pachev), "On the number of classes of integer positive binary quadratic forms whose arithmetical minimum is divided by a given number," Algebra Teoriya Chisel (Nal'chik), 4, 53-67 (1979). r

571

1980

73. "On the application of the discrete ergodic method in the analytic arithmetic of indefinite ternary quadratic forms," Zap. Nauchn. Semin. LOMI, 93, 5-24 (1980). 74. (In collaboration withU. M. Pachev), "On the arithmetic of matrices of the second order," Zap. Nauchn. Semin. LOMI, 93, 41-86 (1980). 1981

75. (In collaboration with N. N. Belova), "Ergodic properties of integer points on ellipsoids of genus ~[ft,1] ," Zap. Nauchn. Semin. LOMI, 106, 17-51 (1981). 1982

76. "Linnik's discrete ergodic method," in: Encyclopedia of Mathematics, Vol. 3, Soviet. Moscow (1982), pp. 389-390. 77. "Ray functions," in: Encyclopedia of Mathematics, Vol. 3, Soviet. Entsiklopediya, p. 463. 78. "Markov's spectrum problem," in: Encyclopedia of Mathematics, Vol. 3, Soviet. Moscow (1982), pp. 514-516. 79. "Markov's forms," in: Encyclopedia of Mathematics., Vol. 3, Soviet. Entsiklopediya, p. 516.

Entsiklopediya, Moscow (1982), Entsiklopediya, Moscow (1982),

1983

80. (In collaboration with N. M. Glazunov), "Minkowski's conjecture concerning the critical determinant of the region [xlV + ly[p < 1," in: Transcendental Number Theory and Its Applications. All-Union Conf., Abstracts, Moscow (1983), pp. 35-36. 81. (In collaboration with Nguyen Ngor Goi), "On the distribution of integer points on some hyperboloids of a sheet," Zap. Nauchn. Semin. LOMI, 121, 83-93 (1983). 1984

82. "On the geometry of numbers," Comments to H. Weyl's book Selected Works in Mathematics. Theoretical Physics, Nauka, Moscow (1984), p. 488. 83. (In collaboration with U. M. Pachev), "Ergodic properties of integer points on some hyperboloids of two sheets," [in Russian] Manuscript deposited at VINITI 10.09.84, deposition number 6127-84 DEP, Moscow (1984). 84. "The discrete ergodic method and its applications to the arithmetic of ternary quadratic forms," in: Topics in Classical Number Theory, Budapest, 1981. 1985

85. (In collaboration with N. M. Glazunov), "On Minkowski's conjecture and the critical determinant," Kibernetika, No. 5, 10-14 (1985). 1986

86. (In collaboration with N. M. Glazunov), "A proof of Minkowski's conjecture on the critical determinant of the region [x[p + [y[P < 1 in a neighborhood of p = 2," Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7,

9-12 (1986). 87. (In collaboration with N. M. Glazunov and A. S. Golovanov), "A proof of Minkowski's conjecture on the critical determinant of the region Ixlp +[y[P < 1," Zap. Nauchn. Semin. LOMI, 151, 40-53 (1986). 88. (In collaboration with A. K. Andriyasyan and I. V. Ii'in), "On the nonhomogeneous Minkowski conjecture. Estimates of the Chebotarev type for large dimensions," [in Russian], Manuscript deposited at VINITI 07.07.86, deposition number 4933-B, Moscow (1986). 89. (In collaboration with A. K. Andriyasyan and I. V. Ii'in), "On the nonhomogeneous Minkowski conjecture. Application of computers to estimates of the Chebotarev type," [in Russian, Manuscript deposited at VINITI 12.11.86, deposition number 7749-B, Moscow (1986). 90. (In collaboration with A. K. Andriyasyan and I. V. II'in), "On the nonhomogeneous Minkowski conjecture. Estimates of the Chebotarev type for small dimensions. 1," [in Russian], Manuscript deposited at VINITI 16.12.86, deposition number 8575-B, Moscow (1986). f

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91. (In collaboration with U. M. Pachev), "On remainder terms in the ergodic theorem for integer points on some two-sheeted hyperboloids," in: Analytic Number Theory, Petrozavodsk (1986), pp. 46-5!. 92. (In collaboration with A. K. Andriyasyan and I. V. II'in), "Application of computers to estimates of the Chebotarev type in the nonhomogeneous Minkowski conjecture," Zap. Nauchn. Serum. L OMI, 151, 7-25 (1986). 1987

93. "On estimates of a nonhomogeneous arithmetic minimum of the product of linear forms," Zap. Nauchn. Serum. LOMI, 160, 138-150 (1987). 94. "Works of Academician A. A. Markov in number theory," in: Andrei A ndreevich Markov, Nauka, Moscow (1987), pp. 141-163. 95. (In collaboration with B. M. Berdikhin et al.), "Nikolai Grigor'evich Chudakov (obituary), " Usp. Mat. Nauk, 42, No. 5 189-190 (1987). 96. (In collaboration with Yu. G. Teterin and B. M. Shirokov), "Ergodic properties of algebraic fields (on Yu. V. Linnik's program of application of the discrete ergodic method)," in" 19th All-Union Algebraic Conf., Abstracts, Part 1, L'vov (1987), p. 172. 1988

97. (In collaboration with T. U. Asanov), "Translation theorem in the nonhomogeneous Minkowski conjecture," in: Analytic Number Theory, Petrozavodsk (1988). 98. (In collaboration with T. U. Asanov), Translation theorem in the nonhomogeneous Minkowski conjecture," in: 9th All-Union Geometric Conf., Abstracts, Kishinev (1988), pp. 22-23. 1989

99. (In collaboration withA. B. Voronetskii, N. M. Glazunov, and A. S. Golovanov), "A proof of the Minkowski conjecture on the critical determinant," in: Constructive Methods and Algorithms in Number Theory. All-Union School, Abstracts, Minsk (1989), pp. 31-32. 100. (In collaboration with M. N. Kubenskii), "Rotations of embeddings in simple central algebras," in: Constructive Methods and Algorithms in Number Theory. All-Union School, Abstracts, Minsk (1989), p. 79. 1990

101. (In collaboration with M. N. Kubenskii), "The theory of rotations in simple central algebras," Acts Arith., 53, No. 5,447-498 (1990). 1991

102. (In collaboration with B. M. Shirokov), "A new proof of the main lemma of the discrete ergodic method for second-order vector matrices," Vestn. LGU, Ser. Mat., Mekh., Astron., No. 2, 34-40 (1991). THE BOOKS EDITED BY A. V. MALYSHEV 1. Investigations on number theory (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 1 (1966). 2. C. Rogers, Packing and Covering, translated into Russian by B. Z. Moroz and O. M. Fomenko, edited by A. V. Malyshev, Mir, Moscow (1968). 3. E. P. Ozhigova, Development of Number Theory zn Russia, edited by A. V. Malushev, Nauka, Lenin-

grad (1972). 4. J.-P. Serre, Cours d'Arithmdtique, translated into Russian by A. I. Skopin , edited by A. V. Malyshev, Mir, Moscow, (1972). 5. Investigations on number theory, 2 (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 33 (1973). 6. H. Diamond and J. Steining, "An elementary proof of the prime number theorem with a remainder term," translated into Russian by E. V. Podsypanin, edited by A. V. Malyshev, Matematika, 17, No. 1, 97-148 (1973). 7. Investigations on number theory, 3 (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 50 (1975). 573

8. Investigations on number theory, 4 (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 67 (1977). 9. Investigations on number theory, 5 (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 82

(1979). 10. Investigations on number theory, 6 (edited by A. V. Malyshev), Zap. Nauchn. Semin. L OMI, 9 3

(1980). 11. Investigations on number theory, 7 (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 106

(1981). 12. B. A. Venkov, Selected papers. Investigations on number theory (edited by A. V. Malyshev), Nauka, Leningrad (1981). 13. J. Cassels, Rational Quadratic Forms, translated into Russian by B. B. Venkov, edited by A. V. Malyshev, Mir, Moscow (1982). 14. Investigations on number theory, 8 (edited by A. V. Malyshev), Zap. Nauchn. Semin. LOMI, 121

(1983). 15. Investigations on number theory, 9 (edited by A. V. Malyshev and Yu. G. Teterin), Zap. Nauchn. Semin. LOMI, 151 (1986). 16. Automorphic Functions and Number Theory) (edited by A. V. Malyshev), Izhevsk (1987).

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