×

Distribution of integral points on certain two-sheeted hyperboloids. (English) Zbl 0487.10040

Summary: Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 93, 87–141 (Russian) (1980; Zbl 0457.10028).

MSC:

11P21 Lattice points in specified regions
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] B. A. Venkov, Elementary Number Theory, Wolters-Noordhoff, Groningen (1970).
[2] A. I. Vinogradov and Yu. V. Linnik, ?Hyperelliptic curves and the least prime quadratic residue,? Dokl. Akad. Nauk SSSR,168, No. 2, 259?261 (1966). · Zbl 0148.02305
[3] A. M. Istamov, ?Asymptotics of second-order integral matrices lying in a given hyperbolic region and belonging to a given residue class,? J. Sov. Math.,19, No. 2 (1982). · Zbl 0487.10041
[4] Yu. V. Linnik, ?On the representations of large numbers by positive, ternary quadratic forms,? Izv. Akad. Nauk SSSR, Ser. Mat.,4, Nos. 4/5, 363?402 (1940).
[5] Yu. V. Linnik, ?Some applications of the Lobachevskii geometry to the theory of binary quadratic forms,? Dokl. Akad. Nauk SSSR,93, No. 6, 973?974 (1953).
[6] Yu. V. Linnik, ?A new arithmetic application of the geometry of Lobachevskii,? Dopov. Akad. Nauk Ukr. RSR, No. 2, 112?114 (1955).
[7] Yu. V. Linnik, ?The asymptotic distribution of reduced binary quadratic forms in relation to the geometries of Lobachevskii. I?III,? Vestn. Leningr. Univ., No. 2, 3?23; No. 5, 3?32; No. 8, 15?27 (1955).
[8] Yu. V. Linnik, ?The asymptotic geometry of the Gaussian genera; an analogue of the ergodic theorem,? Dokl. Akad. Nauk SSSR,108, No. 6, 1018?1021 (1956).
[9] Yu. V. Linnik, ?Other analogies of ergodic theorems for imaginary quadratic fields,? Dokl. Akad. Nauk SSSR,109, No. 4, 694?696 (1956).
[10] Yu. V. Linnik, ?Some applications of non-Euclidean geometry to the theory of Dirichlet characters; analogies to ergodic theorems,? in: Proc. 3rd All-Union Math. Congress, Moscow, 1956, Vol. 3, Izd. Akad. Nauk SSSR (1958), pp. 21?29.
[11] Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Springer-Verlag, New York (1968). · Zbl 0162.06801
[12] Yu. V. Linnik and A. V. Malyshev, ?Applications of quaternion arithmetic to the theory of ternary quadratic forms and to the expansion of numbers into cubes,? Usp. Mat. Nauk,8, No. 5, 3?71 (1953); correction:10, No. 1, 243?244 (1955).
[13] A. V. Malyshev, ?On representations of integers by positive quadratic forms,? Tr. Mosk. Inst. Akad. Nauk, Vol. 65, Moscow-Leningrad (1962).
[14] A. V. Malyshev, ?A new version of the ergodic method of Yu. V. Linnik in number theory,? Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,50, 179?186 (1975).
[15] A. V. Malyshev, ?The discrete ergodic method of Yu. V. Linnik and its subsequent development,? in: Yu. V. Linnik, Selected Works. Theory of Numbers. Ergodic Method and L-Functions, Leningrad (1979), pp. 418?430.
[16] A. V. Malyshev, ?Application of the discrete ergodic method to the analytical arithmetic of indefinite ternary quadratic forms,? J. Sov. Math.,19, No. 2 (1982). · Zbl 0487.10020
[17] A. V. Malyshev and U. M. Pachev, ?On the representation of integers by positive ternary quadratic forms (a new modification of the discrete ergodic method),? Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,82, 33?87 (1979). · Zbl 0425.10024
[18] A. V. Malyshev and U. M. Pachev, ?Arithmetic of second-order matrices,? J. Sov. Math.,19, No. 2 (1982). · Zbl 0487.10047
[19] E. V. Podsypanin, ?Distribution of integral points on the determinant surface,? J. Sov. Math.,19, No. 2 (1982). · Zbl 0487.10043
[20] B. F. Skubenko, ?The asymptotic distribution of integers on a one-sheeted hyperboloid and ergodic theorems,? Izv. Akad. Nauk SSSR, Ser. Mat.,26, No. 5, 721?752 (1962).
[21] P. Bachmann, Die Arithmetik der quadratisehen Formen, Abt. 1, Teubner, Leipzig-Berlin (1898).
[22] Yu. V. Linnik, ?An application of the theory of matrices and of Lobatschevskian geometry to the theory of Dirichlet’s real characters,? J. Indian Math. Soc.,20, Nos. 1?3, 37?45 (1956). · Zbl 0072.03602
[23] Ju. V. Linnik, Sur une application du theoreme d’Andre Weil a la theorie des caracteres de Dirichlet. Seminaire Delange-Pisot-Poitou. Theorie des nombres. 8e annee, No. 6, Paris, 1966/1967 (1968), pp. 01?07.
[24] A. V. Malyshev, ?Yu. V. Linnik’s ergodic method in number theory,? Acta Arith.,27, 555?598 (1975).
[25] G. Pall, ?Representation by quadratic forms,? Can. J. Math.,1, 344?364 (1949). · Zbl 0034.02201 · doi:10.4153/CJM-1949-032-9
[26] H. J. S. Smith, ?On the orders and genera of ternary quadratic forms,? Phil. Trans.,157, 255?298 (1867); reprinted in: Collected Mathematical Papers, Vol. I, Chelsea, New York, pp. 455?509. · doi:10.1098/rstl.1867.0012
[27] G. L. Watson, Integral Quadratic Forms, Cambridge University Press (1960). · Zbl 0090.03103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.