Rodion Kuzmin

Rodion Osievich Kuzmin (

• 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

${\displaystyle x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}}$

is its continued fraction expansion, find a bound for

${\displaystyle \Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),}$

where

${\displaystyle x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.}$

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

${\displaystyle |\Delta _{n}(s)|\leq Ce^{-\alpha {\sqrt {n}}}~,}$

where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7ⁿ by Paul Lévy.

• In 1930, Kuzmin proved
  quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant

${\displaystyle 2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots }$

is transcendental. See Gelfond–Schneider theorem for later developments.

• He is also known for the Kusmin-Landau inequality: If ${\displaystyle f}$ is continuously differentiable with monotonic derivative ${\displaystyle f'}$ satisfying ${\displaystyle \Vert f'(x)\Vert \geq \lambda >0}$ (where ${\displaystyle \Vert \cdot \Vert }$ denotes the
  Nearest integer function
  ) on a finite interval ${\displaystyle I}$, then

${\displaystyle \sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.}$

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.)