Yuri Linnik

Yuri V. Linnik

Yuri Vladimirovich Linnik (Russian: Ю́рий Влади́мирович Ли́нник; 8 January 1915 – 30 June 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics.

Linnik was born in Bila Tserkva, in present-day Ukraine. He went to St Petersburg University where his supervisor was Vladimir Tartakovski, and later worked at that university and the Steklov Institute. He was a member of the Russian Academy of Sciences, as was his father, Vladimir Pavlovich Linnik. He was awarded both State and Lenin Prizes. He died in Leningrad.^[1]

Work in number theory

Linnik's theorem in analytic number theory
The dispersion method (which allowed him to solve the Titchmarsh problem).^[2]
The large sieve (which turned out to be extremely influential).
An elementary proof of the Hilbert-Waring theorem; see also Schnirelmann density.
The Linnik ergodic method, see Linnik (1968), which allowed him to study the distribution properties of the representations of integers by integral ternary quadratic forms.^[3]

Work in probability theory and statistics

Infinitely divisible distributions

Linnik obtained numerous results concerning infinitely divisible distributions.^[4] In particular, he proved the following generalisation of Cramér's theorem: any divisor of a convolution of Gaussian and Poisson random variables is also a convolution of Gaussian and Poisson.

He has also coauthored the book Linnik & Ostrovskii (1977) on the arithmetics of infinitely divisible distributions.

Central limit theorem

Linnik zones (zones of asymptotic normality)
Information-theoretic proof of the central limit theorem

Statistics

Behrens–Fisher problem

Selected publications

Linnik, Yu.V. (1971), Independent and stationary sequences of random variables, Series of Monographs and Textbooks on Pure and Applied Mathematics, Groningen: Wolters-Noordhoff Publishing
Linnik, Yu.V. (1961), Method of least squares and principles of the theory of observations, New York-Oxford-London-Paris: Pergamon Press, MR 0124121
Linnik, Yu.V.; Ostrovskii, I.V. (1977), Decomposition of random variables and vectors, Translations of Mathematical Monographs, 48, Providence, R.I.: American Mathematical Society
Linnik, Yu.V. (1968), Ergodic properties of algebraic fields, Ergebnisse der Mathematik und ihrer Grenzgebiete, 45, New York: Springer-Verlag New York Inc.

Notes

↑ Faddeyev, D. K.; Lozinsky, S. M.; Malyshev, A. V. (1975), "Yuri V. Linnik (1915–1972): a biographical note", Acta Arith., 27: 1&ndash, 2, doi:10.4064/aa-27-1-1-2, MR 0421941.
↑ Bredikhin, B.M. (2001) [1994], "Density method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
↑ Michel, Ph.; Venkatesh, A. (2006), "Equidistribution, L-functions and ergodic theory: on some problems of Yu. V. Linnik", Proceedings of ICM 2006, 2, Zurich: Eur. Math. Soc., pp. 421&ndash, 457
↑ Ibragimov, I. A. (1992), "Yu. V. Linnik. Some of his work from the 1950s", St. Petersburg Math. J., 3 (3): 687&ndash, 696, MR 1150561.

External links

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[1] Faddeyev, D. K.; Lozinsky, S. M.; Malyshev, A. V. (1975), "Yuri V. Linnik (1915–1972): a biographical note", Acta Arith., 27: 1&ndash, 2, doi:10.4064/aa-27-1-1-2, MR 0421941.

[2] Bredikhin, B.M. (2001) [1994], "Density method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

[3] Michel, Ph.; Venkatesh, A. (2006), "Equidistribution, L-functions and ergodic theory: on some problems of Yu. V. Linnik", Proceedings of ICM 2006, 2, Zurich: Eur. Math. Soc., pp. 421&ndash, 457

[4] Ibragimov, I. A. (1992), "Yu. V. Linnik. Some of his work from the 1950s", St. Petersburg Math. J., 3 (3): 687&ndash, 696, MR 1150561.