Rodion Osievich Kuzmin (russian: Родион Осиевич Кузьмин, 9 November 1891, Riabye village in the

Haradok Haradok ( be, Гарадок, - russian: Городок, Gorodok, pl, Horodek) is a town in the Vitebsk Region of Belarus with the population of 34,700 people. Approximately 14,000 people reside in the town itself around 30,000 people reside wit ...

district – 24 March 1949,

Leningrad Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...

) was a Soviet

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known for his works in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

and

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

. His name is sometimes transliterated as Kusmin. He was an Invited Speaker of the ICM in 1928 in Bologna.

Selected results

* In 1928, Kuzmin solved the following problem due to

Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

(see Gauss–Kuzmin distribution): if ''x'' is a random number chosen uniformly in (0, 1), and ::

x = \frac

:is its

continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...

expansion, find a bound for ::

\Delta_n(s) = \mathbb \left\ - \log_2(1+s),

:where ::

x_n = \frac .

:Gauss showed that ''Δ''_''n'' tends to zero as ''n'' goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that ::

, \Delta_n(s),  \leq  C e^~,

:where ''C'',''α'' > 0 are numerical constants. In 1929, the bound was improved to ''C'' 0.7^''n'' by Paul Lévy. * In 1930, Kuzmin proved that numbers of the form ''a''^''b'', where ''a'' is algebraic and ''b'' is a real

quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...

, are transcendental. In particular, this result implies that

Gelfond–Schneider constant The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two: :2 = ... which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently prov ...

2^=2.6651441426902251886502972498731\ldots

:is transcendental. See

Gelfond–Schneider theorem In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement : If ''a'' and ''b'' are ...

for later developments. * He is also known for the Kusmin-Landau inequality: If

f

is continuously differentiable with monotonic derivative

f'

satisfying

\Vert f'(x) \Vert \geq \lambda > 0

(where

\Vert \cdot \Vert

denotes the

Nearest integer function Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to ob ...

) on a finite interval

I

, then ::

\sum_ e^\ll \lambda^.

Notes

External links

* (The chronology there is apparently wrong, since

J. V. Uspensky ) , birth_date = , birth_place = Urga, Outer Mongolia , death_date = , death_place = San Francisco, United States , nationality = , work_institution = Stanford University, University of Minnesota , alma_mater = University o ...

lived in USA from 1929.) {{DEFAULTSORT:Kuzmin, Rodion 1891 births 1949 deaths People from Gorodoksky Uyezd Soviet mathematicians Number theorists Mathematical analysts Academic staff of Perm State University