Rodion Kusmin, circa 1926
9 October 1891|
Riabye village in the Haradok district
|Died||April 23, 1949
|Institutions||Perm State University, Tomsk Polytechnic University, Saint Petersburg State Polytechnical University|
|Alma mater||Saint Petersburg State University nee Petrograd University|
|Doctoral advisor||James Victor Uspensky|
|Known for||Gauss–Kuzmin distribution, number theory and mathematical analysis.|
Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин, Nov. 9, 1891, Riabye village in the Haradok district – March 23, 1949, Leningrad) was a Russian mathematician, known for his works in number theory and analysis. His name is sometimes transliterated as Kusmin.
- In 1928, Kuzmin solved the following problem due to Gauss (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and
- is its continued fraction expansion, find a bound for
- Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that
- where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy.
- In 1930, Kuzmin proved that numbers of the form ab, where a is algebraic and b is a real quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant
- is transcendental. See Gelfond–Schneider theorem for later developments.
- He is also known for the Kusmin-Landau inequality: If is continuously differentiable with monotonic derivative satisfying (where denotes the Nearest integer function) on a finite interval , then