Three problems on multiplicative functions and related sequences.

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.