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. 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 (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and
x = 1 k 1 + 1 k 2 + ⋯ {\displaystyle x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}}
is its continued fraction expansion, find a bound for
Δ n ( s ) = P { x n ≤ s } − log 2 ( 1 + s ) , {\displaystyle \Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),}
where
x n = 1 k n + 1 + 1 k n + 2 + ⋯ . {\displaystyle x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.}
Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that
| Δ n ( s ) | ≤ C e − α n , {\displaystyle |\Delta _{n}(s)|\leq Ce^{-\alpha {\sqrt {n}}}~,}
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
2 2 = 2.6651441426902251886502972498731 … {\displaystyle 2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots }
is transcendental. See Gelfond–Schneider theorem for later developments. He is also known for the Kusmin-Landau inequality: If f {\displaystyle f} is continuously differentiable with monotonic derivative f ′ {\displaystyle f'} satisfying ‖ f ′ ( x ) ‖ ≥ λ > 0 {\displaystyle \Vert f'(x)\Vert \geq \lambda >0} (where ‖ ⋅ ‖ {\displaystyle \Vert \cdot \Vert } denotes the Nearest integer function) on a finite interval I {\displaystyle I} , then
∑ n ∈ I e 2 π i f ( n ) ≪ λ − 1 . {\displaystyle \sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.}
Notes
External links Rodion Kuzmin at the Mathematics Genealogy Project (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.)
