Three problems on multiplicative functions and related sequences.

圆法是调和分析的一种工具，传统上用于研究加法数论问题，如华林的问题，将数字写成k次幂的和。除了在这些应用中不断取得令人兴奋的发展外，圆方法还可以用来研究一些传统上使用其他工具（如Dirichlet L-函数）研究的乘法数论问题。这篇博士论文的第一部分将在这些方向上进行新的研究，在哈珀和Soundararajan最近的工作的基础上，使用圆方法研究算术级数中一些乘法函数的方差。我们通常会研究确定性对象的随机模型，以确定某个属性是否适用于它们。试图了解有限整数集的产品集的大小，这篇博士论文的第二部分将描述那些随机集的selfproduct几乎肯定是最大的，提高工作的Cilleruelo，Ramana和Ramaré，通过扩展研究的功能密切相关的除数函数。在希望扩大我们的知识的Moebius函数，这篇博士论文的第三部分将集中在推导几乎肯定的上界的某些部分和的随机乘法函数。哈珀最近发现了一个相应的Omega结果，他最近对随机乘法函数的全部部分和的低阶矩估计将被用作增加这种界的强度的一个关键工具。这一切都符合EPSRC的数论研究领域。

The circle method is a tool from harmonic analysis that has traditionally been applied to investigate additive number theory problems, such as Waring's problem about writing numbers as a sum of k-th powers. As well as continuing exciting developments in those applications, the circle method can also be used to investigate some multiplicative number theory problems that would traditionally be studied using other tools like Dirichlet L-functions. The first part of this PhD thesis will pursue new research in these directions, investigating the variance of some multiplicative functions in arithmetic progressions using the circle method, building on recent work of Harper and Soundararajan. It is quite usual to look at random models of deterministic objects to determine whether a certain property might hold for them. Trying to understand the size of product sets of finite integer sets, the second part of this PhD thesis will characterize those random sets whose selfproduct is almost surely maximal, improving on work of Cilleruelo, Ramana and Ramaré, by extending the study of a function closely related to the divisor function. In the hope of expanding our knowledge of the Moebius function, the third part of this PhD thesis will focus on deriving almost sure upper bounds for certain partial sums of a random multiplicative function. A corresponding Omega result has been very recently discovered by Harper, whose recent low moments estimates for the full partial sums of random multiplicative functions will be employed as a key tool to increase the strength of such bounds. This all fits in to EPSRC's number theory research area.