# Yuri 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

## 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**: 1–2, doi:10.4064/aa-27-1-1-2, MR 0421941. - Bredikhin, B.M. (2001) [1994], "Density method", in Hazewinkel, Michiel (ed.),
*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. 421–457 - Ibragimov, I. A. (1992), "Yu. V. Linnik. Some of his work from the 1950s",
*St. Petersburg Math. J.*,**3**(3): 687–696, MR 1150561.