On the constant in the Turan-Kubilius inequality.

Abstract

Since Kubilius in 1983 proved that the Turan-Kubilius inequality holds with the constant close to 1.5, it has been conjectured that the inequality holds with the constant 1.5. In this thesis the conjecture is settled positively in the case of strongly additive functions for all sufficiently large x. The key to the proof is a lower bound on a bilinear form. This is obtained by constructing very precise approximations for the lowest eigenvalue and eigenvector using the power method from numerical analysis. For the latter construction precise evaluations of the mean values of many complicated arithmetic functions on prime numbers. The mean values were sought using analytic methods and the method of hyperbola.