# Rodion Kuzmin

Rodion Kuzmin
Rodion Kusmin, circa 1926
Born 9 October 1891
Riabye village in the Haradok district
Died April 23, 1949 (aged 57)
Nationality Russian
Fields Mathematics
Institutions Perm State University, Tomsk Polytechnic University, Saint Petersburg State Polytechnical University
Alma mater Saint Petersburg State University nee Petrograd University
Known for Gauss–Kuzmin distribution, number theory and mathematical analysis.

Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин, Nov. 9, 1891, Riabye village in the Haradok district – March 23, 1949, Leningrad) was a Russian mathematician, known for his works in number theory and analysis.[1] His name is sometimes transliterated as Kusmin.

## Selected results

$x = \frac{1}{k_1 + \frac{1}{k_2 + \cdots}}$
is its continued fraction expansion, find a bound for
$\Delta_n(s) = \mathbb{P} \left\{ x_n \leq s \right\} - \log_2(1+s),$
where
$x_n = \frac{1}{k_{n+1} + \frac{1}{k_{n+2} + \cdots}} .$
Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that
$|\Delta_n(s)| \leq C e^{- \alpha \sqrt{n}}~,$
where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy.
$2^{\sqrt{2}}=2.6651441426902251886502972498731\ldots$
is transcendental. See Gelfond–Schneider theorem for later developments.

## Notes

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