Probabilistic and other methods in multiplicative number theory

近年来，通过引入来自不同领域的思想，特别是概率论和组合学，乘法数论取得了一些进展。这篇博士论文将寻求进一步的这些调查路线，并开发新的。首先，本文希望改进Munsch的一些结果，给出Dirichlet函数的高偶矩的几乎尖锐的界限，引入概率数论和其他矩问题的思想来锐化和推广Munsch的估计。这篇论文将继续讨论其他各种乘法问题，可能包括丢番图近似或筛子问题。这一切都符合EPSRC的“数论”研究领域。

In recent years there have been several advances in multiplicative number theory by introducing ideas from rather different fields, notably probability and combinatorics. This PhD thesis will seek to further some of these lines of investigation, and develop new ones. To begin, the thesis looks to improve on some results of Munsch giving almost sharp bounds for high even moments of Dirichlet theta functions, introducing ideas from probabilistic number theory and from other moment problems to sharpen and generalise Munsch's estimates. It is intended that the thesis will move on to a variety of other multiplicative questions, possibly including Diophantine approximation or sieve problems. This all fits into EPSRC's "Number theory" research area.