Chebyshev

Chebyshev is largely remembered for his investigations in number theory.

In 1847 Chebyshev was appointed to the University of St Petersburg. He became a foreign associate of the Institut de France in 1874 and also of the Royal Society.

His work on prime numbers included the determination of the number of primes not exceeding a given number. He wrote an important book Teoria sravneny on the theory of congruences in 1849.

In 1845 Bertrand conjectured that there was always at least one prime between n and 2n for n > 3. Chebyshev proved Bertrand's conjecture in 1850. Chebyshev also came close to proving the prime number theorem, proving that if pi (n)log n)/n had a limit as n-> infinity then that limit is 1. He was unable to prove, however, that

lim ( pi (n)log n)/n as n-> infinity

exists. The proof of this result was only completed two years after Chebyshev's death by Hadamard and (independently) de la Vallée Poussin.

In his work on integrals he generalised the beta function and examined integrals of the form

integral x ^p (1-x) ^q dx.

Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling. The Chebyshev parallel motion is three linked bars approximating rectilinear motion.

He wrote about many subjects, including probability theory, quadratic forms, orthogonal functions, the theory of integrals, the construction of maps, and the calculation of geometric volumes.