Three problems on multiplicative functions and related sequences.

乘法函数及相关数列的三个问题。

• 批准号: 2104215
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2104215
• 关键词: Three; problems; multiplicative; functions; related

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.

圆法是调和分析中的一种工具，传统上用于研究加法数论问题，例如将数字写为 k 次方之和的 Waring 问题。除了在这些应用中不断取得令人兴奋的发展之外，圆方法还可以用于研究一些乘法数论问题，而这些问题传统上是使用狄利克雷 L 函数等其他工具来研究的。本博士论文的第一部分将在这些方向上进行新的研究，以 Harper 和 Soundararajan 最近的工作为基础，使用圆方法研究算术级数中某些乘法函数的方差。查看确定性对象的随机模型以确定某个属性是否适用于它们是很常见的。为了理解有限整数集的乘积集的大小，本博士论文的第二部分将描述那些自积几乎肯定是最大的随机集，通过扩展与除数函数密切相关的函数的研究，改进了 Cilleruelo、Ramana 和 Ramaré 的工作。为了扩展我们对莫比斯函数的了解，本博士论文的第三部分将集中于推导随机乘法函数的某些部分和的几乎确定的上限。 Harper 最近发现了相应的 Omega 结果，他最近对随机乘法函数的全部部分和的低矩估计将被用作增加此类界限强度的关键工具。这一切都符合 EPSRC 的数论研究领域。