Yuri Wladimirowitsch Linnik ( Russian Юрий Владимирович Линник , born January 8, 1915 in Bila Zerkwa in Ukraine , † June 30, 1972 in St. Petersburg ) was a Russian mathematician who dealt with probability theory , mathematical statistics and analytical number theory.

## Live and act

Linnik was the son of a teacher couple (his father Vladimir Pawlowitsch Linnik later became a member of the Soviet Academy of Sciences, he worked on optics) and began studying physics in St. Petersburg in 1932. But he soon switched to mathematics and graduated in 1938. In 1939/40 he was a platoon leader in the Soviet Army, but was released again in 1940 and received his doctorate in 1940 from the University of Leningrad under Vladimir Abramowitsch Tartakowski ( representation of large numbers by positive ternary quadratic forms ), that is, he received the Russian doctorate (which in the West corresponds to the habilitation). The work was stimulated by lectures by Boris Alexejewitsch Wenkow and led to a problem of the geometry of numbers (counting grid points in an ellipsoid). In 1941 he volunteered in the fighting for the Pulkovo Heights near Leningrad. He went through part of the siege, but was evacuated to Kazan in 1941. As early as 1940 he was in the Leningrad branch of the Steklov Institute , to which he was a professor even after the war. At the same time he was from 1944 mathematics professor at the University of Leningrad .

First he dealt with number theory. In his dissertation he applied methods of ergodic theory (as in Linnik's theorem about the asymptotic distribution of integer points on a sphere with increasing radius) in it. He dealt with the circle method of Hardy and Littlewood and the method of trigonometric sums in number theory after Ivan Matwejewitsch Vinogradow . In 1941 he introduced the “ big sieve ”, which was the starting point for a completely new branch of analytical number theory. The motive was the search for a proof of the conjecture of the smallest non-residual by Vinogradow, which he succeeded in special cases. In 1950 he introduced the dispersion method into additive number theory. He also investigated density theorems for Dirichlet functions. Later he also turned to statistics and probability theory. Among other things, he solved the Behrens-Fischer problem in statistics on the difference between the mean values ​​of two normally distributed quantities (with different variance). ${\ displaystyle L}$

In 1946 he gave a new proof of Vinogradov's theorem and in 1943 he gave a new elementary proof of Waring's problem (that is, without the aid of analysis), illustrated and widely known in the book Three Pearls of Number Theory by Alexander Yakovlevich Chinchin .

Linnik's theorem poses the question of bounds for the smallest prime number p (d) in an arithmetic progression a, a + d…, a + dn (with a, d relatively prime and d <a, n any positive natural number) in Dependence on d. Linnik proved in 1944 with constants , . After Iwaniec and Kowalski, the theorem is one of the greatest achievements of analytic number theory. The upper bounds for L (Linnik's constant) were lowered more and more over time, starting in 1957 from Pan Chengdong at 10,000 up to 2011 on (Triantafyllos Xylouris) Linnik himself did not give any concrete values ​​for c, L. The constants c, L can be effectively calculated. ${\ displaystyle p (d) ${\ displaystyle c \ geq 1}$${\ displaystyle L \ geq 2}$${\ displaystyle L \ leq 5}$

In 1947 he received the Stalin Prize and in 1970 the Lenin Prize . In 1953 he became a corresponding and in 1964 a full member of the Soviet Academy of Sciences . From 1959 (the year of its establishment) to 1965 he was President of the Leningrad Mathematical Society. He was an external member of the Royal Swedish Academy of Sciences (1971) and an honorary member of the London Mathematical Society. In 1970 he was invited speaker at the International Congress of Mathematicians in Nice (Some recent developments in sequential estimation theory), 1958 in Edinburgh (On divisor problems and some related binary additive problems) and 1962 in Stockholm (On similar regions in mathematical statistics).

His students include Alfréd Rényi , who used Linnik's method of the large sieve, Jonas Kubilius , Ildar Abdulowitsch Ibragimow , Abram Mejerowitsch Kagan and Anatoli Andrianow . Linnik's sieve was expanded in 1965 by his student A. Winogradow and independently of Enrico Bombieri .

## Fonts

• with Ildar Ibragimow: Independent and stationary sequences of random variables. Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, 1971.
• The method of least squares in modern representation. Berlin 1961.
• with IV Ostrovsky: Decomposition of random variables and vectors. American Mathematical Society, Providence, Rhode Island, 1977.
• Decomposition of probability distributions. New York 1964.
• The dispersion method in binary additive problems , AMS 1963
• with AM Kagan , SR Rao: Characterization problems in mathematical statistics. New York, 1973.
• Statistical problems with nuisance parameters. AMS 1968.
• Ergodic properties of algebraic fields. New York 1968.
• Leçons sur les problems de Statistique Analytique. Paris 1967.
• with Alexander Gelfond : Elementary methods in the analytic theory of numbers. Oxford 1966.

## literature

• A. Vinogradov: A brief account of the scientific and pedagogical work of Yu. V. Linnik, Journal of Mathematical Sciences, Volume 137, 2006, No. 2