# Rodion Kuzmin

Rodion Kuzmin
Rodion Kusmin, circa 1926
Born9 October 1891
Riabye village in the Haradok district
DiedMarch 24, 1949 (aged 57)
NationalityRussian
Alma materSaint Petersburg State University nee Petrograd University
Known forGauss–Kuzmin distribution, number theory and mathematical analysis.
Scientific career
FieldsMathematics
InstitutionsPerm State University, Tomsk Polytechnic University, Saint Petersburg State Polytechnical University

Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин, Nov. 9, 1891, Riabye village in the Haradok district – March 24, 1949, Leningrad) was a Russian 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

${\displaystyle x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}}$
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 Δn 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.
${\displaystyle 2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots }$
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

1. ^ Venkov, B. A.; Natanson, I. P. "R. O. Kuz'min (1891–1949) (obituary)". Uspekhi Matematicheskikh Nauk. 4 (4): 148&ndash, 155.
2. ^ 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.
3. ^ Kuzmin, R.O. (1928). "On a problem of Gauss". Dokl. Akad. Nauk SSSR: 375&ndash, 380.
4. ^ Kuzmin, R. O. (1930). "On a new class of transcendental numbers". Izvestiya Akademii Nauk SSSR (math.). 7: 585&ndash, 597.