Biography:Rodion Kuzmin
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Short description: Russian mathematician
Rodion Kuzmin | |
|---|---|
 Rodion Kusmin, circa 1926 | |
| Born | Riabye village in the Haradok district |
| Died | 24 March 1949 (aged 57) Leningrad |
| Nationality | Russian |
| Alma mater | Saint Petersburg State University nee Petrograd University |
| Known for | Gauss–Kuzmin distribution, number theory and mathematical analysis. |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Perm State University, Tomsk Polytechnic University, Saint Petersburg State Polytechnical University |
| Doctoral advisor | James Victor Uspensky |
Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин, 9 November 1891, Riabye village in the Haradok district – 24 March 1949, Leningrad) was a Soviet mathematician, known for his works in number theory and analysis.[1] His name is sometimes transliterated as Kusmin. He was an Invited Speaker of the ICM in 1928 in Bologna.[2]
Selected results
- In 1928, Kuzmin solved[3] the following problem due to Gauss (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and
- [math]\displaystyle{ x = \frac{1}{k_1 + \frac{1}{k_2 + \cdots}} }[/math]
- is its continued fraction expansion, find a bound for
- [math]\displaystyle{ \Delta_n(s) = \mathbb{P} \left\{ x_n \leq s \right\} - \log_2(1+s), }[/math]
- where
- [math]\displaystyle{ x_n = \frac{1}{k_{n+1} + \frac{1}{k_{n+2} + \cdots}} . }[/math]
- Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that
- [math]\displaystyle{ |\Delta_n(s)| \leq C e^{- \alpha \sqrt{n}}~, }[/math]
- where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy.
- In 1930, Kuzmin proved[4] 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
- [math]\displaystyle{ 2^{\sqrt{2}}=2.6651441426902251886502972498731\ldots }[/math]
- is transcendental. See Gelfond–Schneider theorem for later developments.
- He is also known for the Kusmin-Landau inequality: If [math]\displaystyle{ f }[/math] is continuously differentiable with monotonic derivative [math]\displaystyle{ f' }[/math] satisfying [math]\displaystyle{ \Vert f'(x) \Vert \geq \lambda \gt 0 }[/math] (where [math]\displaystyle{ \Vert \cdot \Vert }[/math] denotes the Nearest integer function) on a finite interval [math]\displaystyle{ I }[/math], then
- [math]\displaystyle{ \sum_{n\in I} e^{2\pi if(n)}\ll \lambda^{-1}. }[/math]
Notes
- ↑ Venkov, B. A.; Natanson, I. P.. "R. O. Kuz'min (1891–1949) (obituary)". Uspekhi Matematicheskikh Nauk 4 (4): 148–155. http://mi.mathnet.ru/umn8643.
- ↑ Kuzmin, R. "Sur un problème de Gauss." In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, vol. 6, pp. 83–90. 1929.
- ↑ Kuzmin, R.O. (1928). "On a problem of Gauss". Dokl. Akad. Nauk SSSR: 375–380.
- ↑ Kuzmin, R. O. (1930). "On a new class of transcendental numbers". Izvestiya Akademii Nauk SSSR (Math.) 7: 585–597. http://mi.mathnet.ru/eng/izv5316.
External links
- Rodion Kuzmin at the Mathematics Genealogy Project (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.)
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