Rates of convergence in limit theorems of probabilistic number theory

经典的数论问题，如加法和乘法函数的中值定理和中心极限定理，都得到了很好的研究。这个项目的目的是为这些经典定理提供收敛速度。在这样做的过程中，衡量收敛速度的概率思想将是我们调查的基本工具。为了通过概率模型研究数论问题，我们使用了自然数集的Stone-Cech紧化的思想。根据一个初始数论问题和所涉及的一类算术函数，我们得到了一个或和或积的独立项的模型，这些项分别导致概率中的总和理论或鞅理论。应用相应的收敛速度估计和概率论中的其他适当结果，我们得到了数论中类似的收敛速度结果。近似精度的估计将是一个有其自身利益和价值的问题。在收敛速度的概率度量中有L距离、中心极限定理的整体版本、渐近展开、伊布拉吉莫夫-海德方法、Marcinkiewicz-Zygmund大数定律、完全收敛等。这里不仅研究加性算术函数，还研究乘性和q-乘性的算术函数。

Classical problems of numer 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 initail 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.