Education
Saint St. Petersburg State University.
Saint St. Petersburg State University.
His name is sometimes transliterated as Kusmin. 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 C,α > 0 are numerical constants. In 1930, Kuzmin proved that numbers of the form ab, a is algebraic and b is a real quadratic irrational, are transcendental.
He is also known for the Kusmin-Landau inequality: If is continuously differentiable with monotonic derivative satisfying ( denotes the Nearest integer function) on a finite interval, then
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 is its continued fraction expansion, find a bound for where Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that 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 is transcendental. See Gelfond–Schneider theorem for later developments. He is also known for the Kusmin-Landau inequality: If is continuously differentiable with monotonic derivative satisfying (where denotes the Nearest integer function) on a finite interval, then
Academy of Sciences of the Union of the Soviet Socialist Republics.
