{{Short description|Russian mathematician}} {{Infobox scientist | name = Rodion Kuzmin | image =Kusmin.jpg | image_size = 120px | caption = Rodion Kusmin, circa 1926 | birth_date = {{birth date|1891|10|09|df=y}} | birth_place = Riabye village in the Haradok district | death_date = {{death date and age|1949|03|24|1891|10|09|df=y}} | death_place = Leningrad | field = Mathematics | work_institution = 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''' ({{langx|ru|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.<ref>{{cite journal|last1=Venkov|first1=B. A.|last2=Natanson|first2=I. P.|author2-link=Isidor Natanson|title=R. O. Kuz'min (1891–1949) (obituary)|journal=Uspekhi Matematicheskikh Nauk|volume=4|issue=4|pages=148–155|url=http://mi.mathnet.ru/umn8643}}</ref> His name is sometimes transliterated as Kusmin. He was an Invited Speaker of the ICM in 1928 in Bologna.<ref>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.</ref>
==Selected results== * In 1928, Kuzmin solved<ref>{{cite journal|last=Kuzmin|first=R.O.|title=On a problem of Gauss|journal=Dokl. Akad. Nauk SSSR|year=1928|pages=375–380}}</ref> the following problem due to Gauss (see Gauss–Kuzmin distribution): if ''x'' is a random number chosen uniformly in (0, 1), and ::<math> x = \frac{1}{k_1 + \frac{1}{k_2 + \cdots}}</math> :is its continued fraction expansion, find a bound for ::<math> \Delta_n(s) = \mathbb{P} \left\{ x_n \leq s \right\} - \log_2(1+s),</math> :where ::<math> x_n = \frac{1}{k_{n+1} + \frac{1}{k_{n+2} + \cdots}} .</math> :Gauss showed that ''Δ''<sub>''n''</sub> tends to zero as ''n'' goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that ::<math> |\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'' 0.7<sup>''n''</sup> by Paul Lévy.
* In 1930, Kuzmin proved<ref>{{cite journal|last=Kuzmin|first=R. O.|title=On a new class of transcendental numbers|journal=Izvestiya Akademii Nauk SSSR (Math.)|volume=7|year=1930|pages=585–597|url=http://mi.mathnet.ru/eng/izv5316}}</ref> that numbers of the form ''a''<sup>''b''</sup>, where ''a'' is algebraic and ''b'' is a real quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant ::<math>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> f </math> is continuously differentiable with monotonic derivative <math>f'</math> satisfying <math> \Vert f'(x) \Vert \geq \lambda > 0</math> (where <math>\Vert \cdot \Vert </math> denotes the Nearest integer function) on a finite interval <math>I</math>, then ::<math> \sum_{n\in I} e^{2\pi if(n)}\ll \lambda^{-1}. </math>
==Notes== {{Reflist}}
==External links== * {{MathGenealogy|id=152494}} (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.)
{{Authority control}}
{{DEFAULTSORT:Kuzmin, Rodion}} Category:1891 births Category:1949 deaths Category:People from Gorodoksky Uyezd Category:Soviet mathematicians Category:Number theorists Category:Mathematical analysts Category:Academic staff of Perm State University