Yuri Linnik

Yuri V. Linnik

Yuri Vladimirovich Linnik (Russian: Ю́рий Влади́мирович Ли́нник; January 8, 1915 – June 30, 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

1. Faddeyev, D. K.; Lozinsky, S. M.; Malyshev, A. V. (1975), "Yuri V. Linnik (1915–1972): a biographical note", Acta Arith. 27: 1–2, MR 0421941.
2. Bredikhin, B.M. (2001), "Density method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, 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. 421–457
4. Ibragimov, I. A. (1992), "Yu. V. Linnik. Some of his work from the 1950s", St. Petersburg Math. J. 3 (3): 687–696, MR 1150561.

External links

• Yuri Linnik at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Yuri Linnik", MacTutor History of Mathematics archive, University of St Andrews.
• Acta Arithmetica: Linnik memorial issue (1975)
• List of books by Linnik provided by National Library of Australia