Rates of convergence in limit theorems of probabilistic number theory

数论的经典问题，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的结果。在这样做的过程中，测量收敛速率的概率思想将成为我们研究的基本工具。为了通过概率模型来研究数论问题，我们使用了自然数集的斯通-切赫紧化思想。根据一个初始的数论问题，以及涉及到的一类算术函数，我们分别得到一个独立项的和或积的模型，从而得出概率论中的和论或鞅论。应用概率论中相应的收敛速度估计和其他适当的结果，我们得到了数论中类似的收敛速度结果。估计近似值的准确性将是一个有其自身兴趣和价值的问题。收敛速度的概率度量有lsamvy距离、中心极限定理的全局版本、渐近展开式、Ibragimov-Heyde方法、Marcinkiewicz-Zygmund大数定律、完全收敛等。这里不仅要研究加性算术函数，而且要研究乘性和q乘性算术函数。

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.