Rodion Osievich Kuzmin (, 9 November 1891, Riabye village in the Haradok district – 24 March 1949,

Leningrad Saint Petersburg, formerly known as Petrograd and later Leningrad, is the List of cities and towns in Russia by population, second-largest city in Russia after Moscow. It is situated on the Neva, River Neva, at the head of the Gulf of Finland ...

) 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, mathematical structure, structure, space, Mathematica ...

, known for his works in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

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 (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

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

x = \frac

:is its

continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...

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 or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numb ...

, 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 independent ...

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 Comments The values o ...

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) on a finite interval

I

, then ::

\sum_ e^\ll \lambda^.

Notes

External links

* (The chronology there is apparently wrong, since J. V. Uspensky 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