Rates of convergence in limit theorems of probabilistic number theory

• Applicant: Professor Dr. Josef G. Steinebach
• Subject Area: Mathematics
• Term: from 2005 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5450490

Classical problems of numBer theory, such as the mean-value theorem and the central limit theorem for additive and multiplicative functions, are well-studied. It is the aim of this project to provide rates of convergence results for these classical theorems. In doing so, probabilistic ideas of measuring rates of convergence will be essential tools for our investigations. In order to study a number theoretic problem via a probabilistic model, we use the idea of the Stone-Cech compactification of the set of natural numbers. Depending on an initial number theoretic problem, and a class of arithmetic functions involved, we arrive at a modell of either sums or products, respectively, of independent terms leading either to summation theory or to martingale theory in probability. Applying corresponding convergence rates estimates and other appropriate results form probability theory, we get an analogous rate of convergence result in number theory. The estimation of the accuracy of approximations will be a problem having its own interest and value. Among probabilistic measures for the rate of convergence are the Lévy distance, global versions of the central limit theorem, asymptotic expansions, the Ibragimov-Heyde method, the Marcinkiewicz-Zygmund law of large numbers, complete convergence, and others. Not only additive arithmetic functions are to be studied here, but also multiplicative and q-multiplicative ones.

• DFG Programme: Research Grants
• International Connection: Ukraine
• Participating Person: Professor Dr. Oleg I. Klesov