A new version of the ergodic method of Yu. V. Linnik in number theory

A new version of a known method of analytic number theory is developed. This version is demonstrated with a very simple model example — the problem on the asymptotically uniform distribution of integer points of a sphere with respect to a given modulus.This paper is a reworking of §5 of survey [6].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 179–186, 1975.     

On an advice of Yu. V. Linnik, Hecke operators, and theta functions

This is a historical survey of the Hecke operators and their action on the theta functions and the theta series of integral quadratic forms. Bibliography: 33 titles. Translated fromZapiski Nauchnykh Serninarov POMI, Vol. 226, 1996, pp. 5–13.     

A theorem on the distribution of quadratic residues having applications in the ergodic method of Yu. V. Linnik

A theorem is proved which is used in the next paper of the collection. In the paper the following theorem is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 169–178, 1975.I wish to thank A. V. Malyshev for posing the problem and for attention to the work and D. N. Andreev for help in reviewing the paper of A. I. Vinogradov and Yu. V. Linnik.     

On a conjecture of Yu. V. Linnik on the distribution of integer points on a threedimensional sphere

It is shown that the equation

Yu. V. Linnik's conjecture

One obtains a nontrivial estimate for the mean square variant of Yu. V. Linnik's conjecture on Kloosterman's sum of sums.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 91, pp. 94–118, 1979.     

A brief survey of scientific results of E. A. Storozhenko

We present a survey of the scientific results obtained by E. A. Storozhenko and related results of her disciples and give brief information about the seminar on the theory of functions held under her guidance.     

The Linnik conjecture. A local approach

The Linnik sum is expanded with respect to the spectrum by a local method. Bibliography: 9 titles. To the memory of A. I. Skopin (1927–2003) Deceased. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 71–80.     

Rotations of integral matrices (remark on a theorem of YU. V. Linnik)

We provide a proof of a theorem of Yu. V. Linnik in the arithmetic of integral n×n matrices (cf. Yu. V. Linnik, Ergodic Properties of Algebraic Number Fields, Springer-Verlag, Berlin-Heidelberg-New York (1968), p. 154, Theorem VII.2.2), which he stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 169–170, 1983.     

From Implosion to Fractal Spheres:: a brief account of the historical development of scientific ideas leading to the Trinity test and beyond

The present essay gives a short review of the history of the so-called scientific progress leading to the infamous first Trinity test in New Mexico.A brief speculative discussion of the role played by fractal geometry, nonlinear dynamics and quantum technology in the field is given. We conclude with some subjective remarks about the moral dilemma presented by the current international situation.     

Refinements of Yu. V. Prokhorov's theorems on the asymptotic behavior of the binomial distribution

One finds the asymptotic behavior of the minimax distance between the binomial and its approximating normal or Poisson distributions, taken with the second terms of their expansions.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 4–19, 1986.     

On the factorization of probability distribution from Yu. V. Linnik's class L. II

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 48, pp. 3–26, 1987.     

Factorization of probability distributions from Yu. V. Linnik's class L. I

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 47, pp. 3–25, 1987.     

A theorem of Yu. A. Aminov

We give a simple proof of a somewhat stronger theorem of Yu. A. Aminov concerning a lower estimate of the diameter of a surface immersed in R, this estimate being a function of the mean curvature of the surface and of its Ricci curvature. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 45, pp. 53–55, 1974.